text:
User:Egm6321.f10.team5/hw4

= Problem 1 - Solve for L2-ODE-VC = From Meeting 22, p. 22-3

Given

 * {| style="width:100%" border="0"

 * style="width:95%" | $$\displaystyle  F = (cos \ x) y'' + (x^2 - sin \ x)y' + 2xy = 0$$
 * (1.1)
 * }
 * }

Find

 * 1) Show (1.1) is exact
 * 2) Find $$\Phi$$
 * 3) Solve for $$y(x)$$

1. Show (1.1) is exact
1st Condition of Exactness

from Mtg 15 (2) p. 15-2
 * {| style="width:100%" border="0"

 * style="width:95%" | $$\displaystyle F = \underbrace{(cos \ x)}_{f(x,y,p)} y'' + \underbrace{(x^2 - sin \ x)y' + 2xy}_{g(x,y,p)} = 0$$
 * (1.2)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

(1.1) satisfies the 1st Condition of Exactness
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle f \ = \ cos \ x $$

$$\displaystyle f_{xx} \ = \ -cos \ x $$

$$\displaystyle f_{xy} \ = \ f_{yy} \ = \ f_{xp} \ = \ f_{yp} \ = \ f_{y} \ = \ 0 $$

$$\displaystyle g \ = \ (x^2 - sin \ x) p + 2xy $$

$$\displaystyle g_{xp} \ = \ 2x - cos \ x $$

$$\displaystyle g_{y} \ = \ 2x $$

$$\displaystyle g_{yp} \ = \ g_{pp} \ = \ 0 $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \underbrace{f_{xx} + 2pf_{xy} + p^2 f_{yy}}_{-cos \ x} \ = \ \underbrace{g_{xp} + p g_{yp} - g_{y}}_{-cos \ x} $$

$$\displaystyle \underbrace{f_{xp} + p f_{yp} + 2 f_{y}}_{0} \ = \ \underbrace{g_{pp}}_{0} $$

 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

(1.1) satisfies the 2nd Condition of Exactness
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

Therefore, (1.1) is Exact
 * }
 * }
 * }

2. Find $$\Phi$$
from Mtg 17 (1) p. 17-3

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \phi \ = \ h(x,y) + \underbrace{\int f(x,y,p) \ dp}_{p \ cos \ x} $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \phi_{x} \ = \ h_{x} - p \ sin \ x $$

$$\displaystyle \phi_{y} \ = \ h_{y} $$

$$\displaystyle g \ = \ (h_{x} - p \ sin \ x) + h_{y} \ p \ = \ (h_{y} - sin \ x)p + h_{x} $$

 * }
 * }
 * }

Compare to (1-2), we can choose easily

 * $$\displaystyle h_{y} = x^2$$

 * $$\displaystyle h_{x} = 2xy$$

 * $$\displaystyle \therefore h = x^2 y$$

 * {| style="width:100%; border:2px solid #8888aa;"

 * style="width:95%" | $$\phi \ = \ x^2 y + p \ cos \ x  \ = \ \underbrace{k}_{constant}$$
 * (1.3)
 * }
 * }

3. Solve for $$y(x)$$
(1.3) -->

 * {| style="width:100%" border="0"

 * style="width:95%" | $$\displaystyle y' + \frac{x^2 y}{cos \ x} \ = \ \frac{k}{cos \ x}$$
 * (1.4)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$h \ = \ exp \int_{}^{x} \frac{s^2}{cos \ s} \ ds$$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

$$y(x) \ = \ \frac{1}{exp \int_{}^{x} \frac{s^2}{cos \ s} \ ds} \ \int_{}^{x} (exp  \int_{}^{s} \frac{t^2}{cos \ t} \ dt) \ \frac{1}{cos \ s} \ ds$$
 * }
 * }
 * }

Author and proof-reader
[Author]
 * Oh, Sang Min

[proof-reader]
 * Mike Faraone

= Problem 2 - Solve for Bessel equation = From Meeting 24, p. 24-1

Given
We are given the Bessel differential equation,
 * $$ \displaystyle F(x,y,y',y) = x^2y + xy' + (x^2-\nu^2)y,  ~\nu \in\mathbb{R}  \! $$.

Find
We must first verify exactness of the given equation using 2 methods: If the given equation is not exact, we must determine whether it can be made exact using integrating factor method (IFM) with $$h(x,y) = x^my^n$$.
 * 1) Relations (1) and (2) p. 15-3 (Meeting 15)
 * 2) Equation (3) p. 22-4 (Meeting 22)

Part 1
The first condition of exactness is satisfied since the Bessel differential equation is in the form
 * {| style="width:100%" border="0"

where,
 * style="width:95%" | $$\displaystyle F(x,y,y',y) = g(x,y,y') + f(x,y,y')y$$
 * (2.1)
 * }
 * }
 * $$\displaystyle g(x,y,y') = xy' + (x^2 - \nu^2)y \! $$
 * $$ f(x,y,y') = x^2 \! $$

The second condition of exactness from p. 15-3 has two relations given here.
 * {| style="width:100%" border="0"

 * style="width:95%" | 1. $$\displaystyle f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_y$$
 * (2.2a)
 * }
 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

where
 * style="width:95%" | 2. $$\displaystyle f_{xp} + pf_{yp} + 2f_y = g_{pp}$$
 * (2.2b)
 * }
 * }
 * $$\displaystyle p := y'$$

Computing the partial derivatives for (3.2a) and (3.2b) yeilds,
 * $$\displaystyle f_{xx} = 2$$
 * $$\displaystyle f_{xy} = f_{yy} = f_{xp} = f_{yp} = f_{y} = 0$$
 * $$\displaystyle g_{xp} = 1$$
 * $$\displaystyle g_{yp} = g_{pp} = 0$$
 * $$\displaystyle g_{y} = x^2 - \nu^2$$

Now we substitute the derivatives into (2.2a) and (2.2b) to get
 * (2.2a) $$\displaystyle \Rightarrow -1 = x^2 - \nu^2 \! $$
 * (2.2b) $$\displaystyle \Rightarrow 0 = 0 \! $$

Clearly, the first relation (2.2a) indicates that the Bessel differential equation is not exact for all $$x$$. Alternatively, the second condition of exactness can also be verified using (3) p.22-4 which is as follows,
 * {| style="width:100%" border="0"

where,
 * style="width:95%" | $$\displaystyle f_0 - \frac{df_1}{dx} + \frac{d^2f_2}{dx^2} = 0$$
 * (2.3)
 * }
 * }
 * $$\displaystyle f_i:=\frac{\partial F}{\partial y^{(i)}}, ~i=0,1,2,...,n$$

For the given equation,
 * $$ f_{0} = \frac{\partial F}{\partial y^{(0)}} = x^2 - \nu^2$$
 * $$ f_{1} = \frac{\partial F}{\partial y^{(1)}} = x ~\Rightarrow ~\frac{df_1}{dx} = 1$$
 * $$ f_{2} = \frac{\partial F}{\partial y^{(2)}} = x^2 ~\Rightarrow ~\frac{d^2f_2}{dx^2} = 2$$

Substituting these results into (2.3) yields,
 * $$\displaystyle x^2 - \nu^2 - 1 + 2 = 0$$

which is the same result obtained from (2.2a) above. Again, the Bessel equation is not exact for all $$x$$.

Part 2
We will now employ the integrating factor method using $$h(x,y) = x^my^n$$ to see if the given equation can be made exact.
 * $$ h(x,y)\Big(x^2y'' + xy' + (x^2-\nu^2)y \Big) = 0 \! $$
 * $$ x^my^n \Big(x^2y'' + xp + (x^2-\nu^2)y \Big) = 0 \! $$

Using the form of (2.1) yields,
 * $$ f(x,y,p) = x^{m+2}y^n \! $$
 * $$ g(x,y,p) = x^{m+1}y^np + x^{m+2}y^{n+1} - x^my^{n+1}\nu^2 = 0 \! $$

To satisfy the 2nd condition of exactness, $$f(x,y,p)$$ and $$g(x,y,p)$$ must satisfy (2.2a) and (2.2b). The partial derivatives for these 2 equations are;
 * $$ f_{xx} = (m+1)(m+2)x^m y^n \! $$
 * $$ f_{xy} = n(m+2)x^{m+1} y^{n-1} \! $$
 * $$ f_{y} = nx^{m+2} y^{n-1} \! $$
 * $$ f_{yy} = n(n-1)x^{m+2} y^{n-2} \! $$
 * $$ f_{xp} = f_{yp} = 0 \! $$
 * $$ g_{xp} = (m+1)x^m y^n \! $$
 * $$ g_{y} = nx^{m+1}y^{n-1}p + (n+1)x^{m+2}y^n - (n+1)x^m y^n \nu^2 \! $$
 * $$ g_{yp} = nx^{m+1}y^{n-1} \! $$
 * $$ g_{pp} = 0 \! $$

Substituting these partial derivatives into (2.2a) and (2.2b) yields
 * (2.2a) $$\displaystyle \Rightarrow (m+1)(m+2)x^my^n + 2pn(m+2)x^{m+1}y^{n-1} + p^2n(n-1)x^{m+2}y^{n-2}$$
 * $$=\displaystyle (m+1)x^my^n + pnx^{m+1}y^{n-1} - npx^{m+1}y^{n-1} -(n+1)x^{m+2}y^n + (n+1)x^my^n\nu^2$$
 * (2.2b) $$\displaystyle \Rightarrow 0 + 0 + 2nx^{m+2}y^{n-1} = 0 \! $$

The second relation (2.2b) ⇒ $$n = 0$$. Therefore, the first relation (2.2a) simplifies to:
 * {| style="width:100%" border="0"

This relation (2.4) cannot be satisfied with any value of $$m$$ to make the Bessel differential equation exact.
 * style="width:95%" | $$\displaystyle (m+1)(m+2)x^m = (m+1)x^m -x^{m+2} + x^m\nu^2$$
 * (2.4)
 * }
 * }

Author and proof-reader
[Author]

 * Michael Steinberg

[proof-reader]

 * Raul Riveros

 * Rob Carroll

 * Mike Faraone

= Problem 3 - Find $$X(x)$$= From Meeting 25, p. 25-2

Given
The following identities are given,
 * $$\displaystyle \exp (i \theta) = \cos (\theta) + i \sin (\theta)$$

and
 * $$\displaystyle \exp (\theta) = \cosh (\theta) + \sinh (\theta)$$.

Find
Find $$X(x)$$ in terms of $$\sin$$, $$\cos$$, $$\sinh$$, and $$\cosh$$.

Solve
Assume that $$X(x) = \exp(rx)$$. From the equation of $$K^4 X^{(4)} = X$$, it was already derived in the lecture notes that
 * $$r_{1,2} = \pm \frac 1 K$$

and
 * $$r_{3,4} = \pm \frac i K$$.

Since there are four solutions for $$r$$, we make a linear combination to express $$X(x)$$.
 * {| style="width:100%" border="0"

 * style="width:95%" | $$\displaystyle X(x) = c_1 \exp (r_1 x) + c_2 \exp (r_2 x) + c_3 \exp (r_3 x) + c_4 \exp (r_4 x)$$
 * (3.1)
 * }
 * }

Substitute the values of $$r$$s into the equation and we get
 * {| style="width:100%" border="0"

 * style="width:95%" | $$\displaystyle X(x) = c_1 \exp (\frac x K) + c_2 \exp (- \frac x K) + c_3 \exp (i \frac x K) + c_4 \exp (- i \frac x K)$$
 * (3.2)
 * }
 * }

Now use the expression of exponential in terms of trigonometric functions.
 * {| style="width:100%" border="0"

 * style="width:95%" | $$\displaystyle X(x) = c_1 \left[\cosh(\frac x K) + \sinh (\frac x K)\right] + c_2 \left[\cosh(- \frac x K) + \sinh (- \frac x K)\right]+ c_3 \left[ \cos (\frac x K) + i \sin (\frac x K) \right] + c_4 \left[ \cos (- \frac x K) + i \sin (- \frac x K) \right]$$
 * (3.3)
 * }
 * }

From the trigonometric identity, it is defined that $$\sin (- \theta) = - \sin (\theta)$$, $$\cos (- \theta) = \cos (\theta)$$, $$\sinh (- \theta) = - \sinh (\theta)$$, and $$\cosh (- \theta) = \cosh (\theta)$$. Apply the relations to the equation of $$X(x)$$.
 * {| style="width:100%" border="0"

Then the equation can be simplified,
 * style="width:95%" | $$\displaystyle X(x) = c_1 \left[\cosh(\frac x K) + \sinh (\frac x K)\right] + c_2 \left[\cosh(\frac x K) - \sinh (\frac x K)\right]+ c_3 \left[ \cos (\frac x K) + i \sin (\frac x K) \right] + c_4 \left[ \cos (\frac x K) - i \sin (\frac x K) \right]$$
 * (3.4)
 * }
 * }
 * {| style="width:100%; border:2px solid #8888aa;"

 * style="width:95%" | $$\displaystyle X(x) = [c_1 + c_2] \cosh(\frac x K) + [c_1 - c_2] \sinh (\frac x K) + [c_3 + c_4] \cos (\frac x K) + [c_3 - c_4] i \sin (\frac x K)$$
 * (3.5)
 * }
 * }

Author and proof-reader
[Author]
 * Jo

[proof-reader]
 * Raul Riveros

= References =

= Contributing members = Mike Faraone: Solved and Proofed Problems 1 & 2.

Jo: Authored Problem 3.

Rob Carroll: Solved and proof read problem 2

Michael Steinberg: Solved problem 2.

Raul Riveros: Proofed problems 2 and 3, edited formatting for all problems.

title: User:Egm6321.f10.team5/hw4


text:
User talk:Egm6321.f10.team5.oh/hw4

= Problem 1 - Solve for L2-ODE-VC = From Meeting 22, p. 22-3

Given

 * {| style="width:100%" border="0" align="left"

$$\displaystyle F = (cos \ x) y'' + (x^2 - sin \ x)y' + 2xy = 0 $$ $$\displaystyle (1-1) $$
 * }
 * }

Find

 * {| style="width:100%" border="0" align="left"

 (1) Show (1-1) is exact

 (2) Find  $$\displaystyle \phi $$

 (3) Solve for  $$\displaystyle y(x) $$
 * }
 * }
 * }

Solve
''' Part 1. Show (1-1) is exact '''

1st Condition of Exactness

from Mtg 15 (2) p. 15-2
 * {| style="width:100%" border="0" align="left"

$$\displaystyle F = \underbrace{(cos \ x)}_{f(x,y,p)} y'' + \underbrace{(x^2 - sin \ x)y' + 2xy}_{g(x,y,p)} = 0 $$

$$\displaystyle (1-2) $$

 * }
 * }

 * {| style="width:100%" border="0" align="left"

(1-1) is satisfied with 1st Condition of Exactness
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle f \ = \ cos \ x $$

$$\displaystyle f_{xx} \ = \ -cos \ x $$

$$\displaystyle f_{xy} \ = \ f_{yy} \ = \ f_{xp} \ = \ f_{yp} \ = \ f_{y} \ = \ 0 $$

$$\displaystyle g \ = \ (x^2 - sin \ x) p + 2xy $$

$$\displaystyle g_{xp} \ = \ 2x - cos \ x $$

$$\displaystyle g_{y} \ = \ 2x $$

$$\displaystyle g_{yp} \ = \ g_{pp} \ = \ 0 $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \underbrace{f_{xx} + 2pf_{xy} + p^2 f_{yy}}_{-cos \ x} \ = \ \underbrace{g_{xp} + p g_{yp} - g_{y}}_{-cos \ x} $$

$$\displaystyle \underbrace{f_{xp} + p f_{yp} + 2 f_{y}}_{0} \ = \ \underbrace{g_{pp}}_{0} $$

 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

(1-1) is satisfied with 2nd Condition of Exactness
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

Therefore, (1-1) is Exact
 * }
 * }
 * }

from Mtg 17 (1) p. 17-3

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \phi \ = \ h(x,y) + \underbrace{\int f(x,y,p) \ dp}_{p \ cos \ x} $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \phi_{x} \ = \ h_{x} - p \ sin \ x $$

$$\displaystyle \phi_{y} \ = \ h_{y} $$

$$\displaystyle g \ = \ (h_{x} - p \ sin \ x) + h_{y} \ p \ = \ (h_{y} - sin \ x)p + h_{x} $$

 * }
 * }
 * }

Compare to (1-2), we can choose easily

 * {| style="width:100%" border="0" align="left"

$$\displaystyle h_{y} = x^2 $$

$$\displaystyle h_{x} = 2xy $$

$$\displaystyle \therefore h = x^2 y $$

 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\phi \ = \ x^2 y + p \ cos \ x \ = \ \underbrace{k}_{constant}$$ $$\displaystyle (1-3) $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }
 * }

(1 - 3) -->
 * {| style="width:100%" border="0" align="left"

 * $$y' + \frac{x^2 y}{cos \ x} \ = \ \frac{k}{cos \ x}$$
 * $$y' + \frac{x^2 y}{cos \ x} \ = \ \frac{k}{cos \ x}$$

$$N_{x} \neq M_{y}$$ $$\displaystyle (1-4) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$h \ = \ exp \int_{}^{x} \frac{s^2}{cos \ s} \ ds$$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

$$y(x) \ = \ \frac{1}{exp \int_{}^{x} \frac{s^2}{cos \ s} \ ds} \ \int_{}^{x} (exp  \int_{}^{s} \frac{t^2}{cos \ t} \ dt) \ \frac{1}{cos \ s} \ ds$$
 * }
 * }
 * }

Author and proof-reader
[Author] Oh, Sang Min

[proof-reader]

= Problem 2 - Solve for Bessel equation = From Meeting 24, p. 24-1

Author and proof-reader
[Author]

[proof-reader]

= Problem 3 - Find X(x) = From Meeting 25, p. 25-2

Author and proof-reader
[Author]

[proof-reader]

= References =

= Contributing members =

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Egm6321.f10.team5.oh/hw4


text:
User talk:Norton~enwikiversity

 Hello Norton, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 19:48, 16 October 2010 (UTC)

Wikistution
You may be surprised to learn that your learning project has been completely imported locally now. -- dark lama  14:49, 17 October 2010 (UTC)
 * Yes, I am surprised (and impressed)! Thank you for your time and help!  Norton 14:56, 17 October 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Norton. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Norton~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:40, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

Removal of a project you helped create
Dear ,

The project Wikistution has been moved to User:Norton~enwikiversity/Wikistution

Yours truly, --Guy vandegrift (discuss • contribs) 19:22, 6 December 2022 (UTC)

This project had subpages:

Guy vandegrift (discuss • contribs) 20:01, 6 December 2022 (UTC)

title: User talk:Norton~enwikiversity


text:
User talk:OneAmongBillions

 Hello OneAmongBillions, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 19:48, 16 October 2010 (UTC)

title: User talk:OneAmongBillions


text:
User:Srhat/vector.css

@import "http://meta.wikimedia.org/w/index.php?title=User:Srhat/global.css&action=raw&ctype=text/css";

title: User:Srhat/vector.css


text:
User:Erwin/vector.js

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title: User:Erwin/vector.js


text:
Motivation and emotion/Textbook/Motivation/Flow

"Imagine that you are Skiing down a slop and your full attention is focused on the movements of your body, the position of the skis, the air whistling past your face, and the snow-shrouded trees running by. There is no room in your awareness for conflicts or contradictions; you know that a distracting thought or emotion might get you buried face down in the snow. The run is so perfect that you want it to last forever"(Csikszentmihalyi, 1997).

Chapter Overview
This chapter examines the motivational topic of flow theory. Readers will learn how this mini-theory is incorporated into the motivation of learning and challenge and why flow is an important positive psychology topic. Students will also learn about Mihaly Csikszentmihalyi, the founder and profound psychologist and researcher who has dedicated many years to the study of flow. In fact, much of the research focused on will be research conducted by either Csikszentmihalyi or his colleagues. By the end of the chapter, students should also begin to understand the applications and practicality of flow theory. Finally, a revision test is presented for those students wishing to enhance their memory on the main keypoints found in this chapter.

Flow Theory and Intrinsic motivation
'How and why is the experience of Flow a topic of Motivation? '

Intrinsic motivation refers to the source of energy that elicits an individual to perform an activity because the activity is interesting and satisfying in itself. This is opposed to doing an activity to obtain an external goal, also known as extrinsic motivation (Reeve, 2009). Different types of motivations have been described based on the degree they are internalised, that is, the ability to transform an extrinsic motive into personally endorsed values (Ryan, 1995). Flow is believed to represent the ultimate form of motivation in regards to internalization and intrinsic motivation (Csikszentmihalyi, 1990). This is perhaps the reason that continuing research has been carried out on flow, to understand the dynamics of the concept so that more people can harness it, and in turn their intrinsic motivation.

Intrinsic motivation is essential to well being because it enables an individual to fulfil the three innate psychological needs. These needs include the need for competence, autonomy and relatedness and they form the basis of a model known as Self Determination Theory (SDT) as proposed by Deci and Ryan (1991). Fulfilment of the three basic needs is associated with vital, agentic behaviour which is characterised by intrinsically motivated, self-regulated, curious, proactive and intentional engagement in activities. SDT examines the choices people make without any external influence or interference and the degree to which an individual’s behaviour is self-motivated and self-determined (Deci & Ryan, 2002). Satisfaction with and pursuit of goal-directed need fulfilment can enhance activity engagement and personal growth (Ryan & Deci, 2000).

SDT and Flow theory bear many of the same principles and this is perhaps why they are often examined together. However, there are differences. SDT is a larger theory, almost alike to intrinsic motivation in general which encompasses all three psychological needs. Flow theory is a mini theory and is it most commonly associated with one psychological need, the need for competence. Competence was originally described by White (1959) who believed a child's primary motive is perceived to be fulfilment of the need for competence or mastery of the environment. This is important because one of the largest principles in Flow is associated with mastery of the task at hand. Competence is beneficial to well-being as it allows one's self efficacy to increase, so harnessing competence through the use of Flow is an excellent way to increase personal growth (Shernoff D., Csikszentmihalyi, Shneider & Shernoff E., 2003).

State of Flow, primarily thought of as a motivational topic also demonstrates many associations and relations with emotion. During flow, the emotions are energized and aligned with the task at hand. Flow will not be experienced if the current emotions are ones of depression or anxiety. If flow is experienced the emotions will be ones of joy and this may explain why flow is commonly thought of as a positive psychology topic (Goleman, 2000). This ability of flow to harness the emotions further demonstrates why it is a major source of intrinsic motivation due to the connectivity that exists between emotion and internalization. Perhaps it is emotion that explains why the state of Flow is such a powerful form of intrinsic motivation, as the positive affect encourages a person to engage further in activities that elicit such emotions through Flow. Schweinle, Meyer & Turner (2006) found that motivation and affect are experienced together, suggesting that learning is not merely a cognitive activity but that it is emotionally stimulating. Also found was a relation between affect and students perception of the importance of a task. This research supports the contention of emotion theorists that affect is inextricably linked to motivation.

Mihaly Csikszentmihalyi
'Who is the founder and what contributions have they made to Flow Research? '

Mihaly Csikszentmihalyi is known as the founder and creator of the theory of flow, having spent many years researching the topic. Flow was first described by Csikszentmihalyi after a number of studies were carried out to determine the conditions that create enjoyment (Csikszentmihalyi, 1975, 1982, 1990). Csikszentmihalyi interviewed hundreds of people he presumed knew what it felt like to have fun such as dancers, rock-climbers, chess champions, basketball players, surgeons and others. In all samples, Csikszentmihalyi found the essence of enjoyment could be traced to the 'flow experience'. Furthermore, the descriptions given of flow did not vary substantially by culture, gender or age, suggesting that state flow is a universal experience (Csikszentmihalyi, 1997).

These initial studies allowed Csikszentmihalyi to develop the concept of flow but with further work he aimed instead on its applications. He is particularly interested in enhancing flow in children through the use of improved education. His emphasis however is on improving the quality of flow rather than the quantity as flow can be attained from activities that are counterproductive from a social point of view (Whalen, 1999). As well as flow, Csikszentmihalyi is also known for work in the study of happiness and creativity and has been described by a former president of the American Psychological Association as being the world's leading researcher on positive psychology. His works are influential and widely cited, with the majority of current research on Flow having been conducted by him and his colleagues (Whalen, 1999).

Here is a video where Mihaly Csikszentmihalyi speaks on his research into Flow:

Components of Flow
What are the characteristics that define the state of Flow?

Flow is the mechanism of completely engrossing oneself in a task without making the conscious decision to do so. Awareness of all other things is lost, including time, distractions and to an extent, even bodily needs. This is believed to occur because all of the attention of the person is concentrated on the task at hand, therefore there is no more attention to be allocated (Csikszentmihalyi M. & Csikszentmihalyi I., 1988). Csikszentmihalyi originally identified nine factors that accompany the experience of flow. He later identified a tenth factor which is also presented here (Csikszentmihalyi, 1997).

 * 1) 	There should be a balance between the individual's ability/skill level and the level of difficulty in the challenge. If the challenge is either too easy or to difficult, then uncomfortable emotions are experienced and flow does not occur. Optimal Skill/Challenge level creates the affect of competency.
 * 2) 	Goals should be clear. That is, expectations are foreseen and goals are attainable. Also, Challenge level and skill level should both be high.
 * 3) 	There should be a high degree of concentration on a limited field of attention. That is, the person involved in the activity will be able to focus and be deeply engaged in the activity.
 * 4) 	A loss of feeling of self-consciousness is experienced.
 * 5) 	Sense of time transcendence. That is, an individual's subjective experience of time is altered. It is a pleasurable state that requires no external regulation.
 * 6) 	Direct and immediate feedback should be available so that behaviour can be adjusted as needed. This creates a merging of action and awareness.
 * 7) 	The individual should hold a sense of personal control over the situation or activity.
 * 8) 	The activity must be intrinsically rewarding, resulting in an effortlessness of action
 * 9) 	Absorption in the activity is essential. Focus of awareness should be narrowed down to the activity itself.
 * 10) 	Lack of awareness of bodily needs. That is, the extent that one can reach a point of great hunger or fatigue without realizing it.

Conditions for Flow
What requirements are needed to experience Flow? There are three conditions necessary to enter and achieve flow:
 * 1) 	One must have a clear set of goals in regards to the activity, thereby adding direction and structure to the task.
 * 2) 	They must also have a good balance between the perceived challenges of the task and his or her own perceived skills.
 * 3) 	The task must have immediate and clear feedback. This enables the person to adjust his or her performance with respect to the changing demands of the task. (Csikszentmihalyi, Abuhamdeh & Nakamura, 2005).

Flow is not predictable nor is it possible to manipulate or be forced. A flow state can be entered while performing any activity, however this is most likely to occur when the task is being performed for intrinsic purposes (Csikszentmihalyi, 1988. Snyder & Lopez, 2007).

In 1975 Csikszentmihalyi published the graph shown to the right. It depicts the state of flow as a balance between challenge and skill. Above flow where challenge starts to increase while skill level remains low, anxiety is experienced. Underneath flow, where skill level is high but challenge is low, Boredom is experienced. This diagram is a very simplified one and it signifies that flow can be experienced provided personal skill and challenge are balanced. However Csikszentmihalyi later noted that the balance between skill and challenge must be moderately high to high in order to enter flow.

In 1977 he published another graph, illustrating that skill and challenge must both be above average (as seen on the left). The centre of the graph represents one's average levels of challenge and skill (Csikszentmihalyi, 1988). There are eight different mental state's one may experience, each identifiable by the emotions that accompany it (Snyder & Lopez, 2007).

In the study by Moneta and Csikszentmihalyi (1996) the effects that perceived challenges and skills elicit on everyday life experience were examined. The study used an ESM method which was administered to students at random times and examined four different domains within four different contexts. The domains measured state of mind within an activity. Firstly, it was found that the balance of challenges and skills has a positive and independent effect on the quality of experience within the domains of concentration and involvement. However, challenge had a negative relationship when looking at the domains of happiness and wanting to do the activity. Secondly, it was found that the balance of challenges and skills enhances quality of experience within all four dimensions. Alterations of these findings were also seen when context of environment such as school or home were accounted for. Behaviour may differ within contexts due to the challenges and skills in each context differing in nature. Within every context, the dimensions of concentration, involvement and happiness reach their maximum values when challenges and skill are both highest (Moneta & Csikszentmihalyi, 1996). Schweinle, Meyer & Turner (2006) found that optimal challenge is necessary in educating students. Students in the sample were examined while studying mathematics because students often perceive it as difficult but valued, perfect for creating Flow conditions. In classrooms perceived as average regarding challenge, students reported high importance. However if challenge was too high or too low, students reported low importance. Where challenge is high, students may devalue the task as a protective mechanism and where low, students may have determined that the material was not important enough to require much effort. These findings support the need for optimal challenge (Schweinle, Meyer & Turner, 2006).

Flow in other cultures
Does the model of Flow differ between cultures?

Flow has been described throughout history and across cultures. The Tibetan Buddhist monks speak of a state of mind similar to flow and many of their teachings describe ways of control the flow of items being attended to and accessing consciousness through meditation (Carter et.al, 2005).

The study by Moneta (2004) hypothesized that for Chinese students the optimal challenge/skill ratio is biased toward skills, and that the bias is stronger for those Chinese who have a more collectivist model of the self. The ESM was used was used to compare Chinese and U.S. students. Findings demonstrated that the optimal challenge/skill ratio is balanced towards skills and that this bias is markedly stronger for the Chinese sample. Chinese tend to experience the highest level of intrinsic motivation through mastery practice (low challenge/high skill) conditions than in flow conditions (High challenge/High skill). Perhaps these results are explained by internalization of collectivist values. Furthermore, for Chinese participants, the independent effect of skills is less negative and the effect of the imbalance of challenges and skills is also less negative. Taken together, these findings indicate that the flow model fits well for Chinese. When examining the disposition of participants it was found that trait intrinsic motivation was the most powerful moderator of the flow model and that the optimal challenge/skill ration for these participants was unbiased. The flow model appeared to be homogenous across U.S. ethnic groups and there was no evidence that Asian-Americans differ from other Americans. Perhaps acculturation may shape an individual's flow model. This research provides supportive evidence that the flow model in everyday life is moderated by culture and dispositions. It also suggests that a multi-cultural development of flow theory may be required (Moneta, 2004)

Personality & Flow
Are some personality traits more conductive to state Flow than others?

Various studies have indicated that certain personality traits may be more suitable in allowing an individual to enter and experience flow. In a study by Wong & Csikszentmihalyi (1991) personality was examined in relation to experience during studying and academic performance. Results found that controlling for ability, Personality was a better indicator of grade than experience. Students who have a tendency to aspire to accomplish difficult tasks and maintain high standards, a need to be organized, and an ability to control impulses are more likely to succeed in school. These students tend to be less self-conscious when they study and achieve better grades. Intrinsic motivation while studying however appears to have a strong relation with the difficulty level of the courses students take. These students probably welcome new challenge, persist in the face of obstacles and maintain their interest in spite of failure. These characteristics are important for long-term achievement. These findings suggest the notion that there are two kinds of motivation in scholastic achievement, one directed towards long term-goals and the other towards on-going experience (Wong & Csikszentmihalyi, 1991). Students with a high work orientation are more likely to be less self-conscious while studying and would probably study more yet they not need necessarily feel happy, motivated or satisfied about their performance while they study. Therefore they tend to study to achieve long term goals such as good grades.

Csikszentmihalyi hypothesized that some people possess specific personality traits ideal to achieving flow. These personality traits may include persistence, curiosity, low self-centeredness, and a high rate of performing activities for intrinsic reasons only, also known as trait intrinsic motivation. People who possess most of these personality traits are said to have an Autotelic personality (Snyder & Lopez, 2007). Asakawa (2009) examined a sample of autotelic Japanese students to examine how they differed from the common population. His results found that autotelic Japanese students or those who experienced flow more often in their daily lives, were more likely to show higher self-esteem and lower anxiety, better use of coping strategies more often and use passive coping strategies less often compared to their less autotelic counterparts. They were also more likely to report active commitments to college life, search for a future career and everyday general activities. In addition, they reported more Jujitsu-Kan, that being a sense of fulfilment and a greater sense of satisfaction with their lives (Asakawa, 2009).

Other studies have also examined personality traits through the use of other personality dimensions. The study by Vitterso (2003) examined the big five personality trait of openness in regards to its effects on flow. The authors firstly acknowledge that unlike Csikszentmihalyi's autotelic concept, openness is not directly connected to flow however they predict that it may elicit or facilitate flow. This prediction is based on the reasoning that possessing higher openness allows people to experience more positivity and more likelihood of approaching challenging tasks. Participants were confronted with cartoons depicting a challenging situation and asked to explain the cartoon's experience. It was found that those individuals higher in openness described the cartoons experience as more positive. In addition, findings revealed that openness to experience had a moderate influence on the satisfaction with life measure (Vitterso, 2003). This research suggests that individuals higher in openness and positivity are more likely to approach challenging situations, thereby creating situations to facilitate flow.

Environment and Flow
Does context count? The environment a person is in can produce highly significant effects on cognitive and affective states. This is important in determining what environments people can achieve flow in, with individuals differing in respect to this. The study by Prescott, Csikszentmihalyi & Graef (1982) used ESM techniques to examine the environmental effects on cognitive and affective states within a sample of adult professionals. Significant contrasts appeared particularly between home and work but also in recreation and transportation as well. Observations from younger respondents indicated that recreation is more positive overall than it was for older participants. Younger participants also demonstrated less pleasure while in transportation and found home more relaxing. A similar study by Csikszentmihalyi & LeFevre (1989) also examined differences in experience and Flow. The findings of this study demonstrated that the majority of flowlike experiences seem to come from work, not leisure, and this is true for upper as well as lower level jobs. Other than work the next greatest amount of flow came surprisingly from driving and the third next, from talking to friends and family (Csikszentmihalyi & LeFevre, 1989). Other research suggests that that feeling close or united with others may be an important conductor to creating flow (Kowal & Fortier, 1999).

Whether work is obligatory or volunteered may make a difference in how a person responds to an activity. This is what the study by Csikszentmihalyi & Figurski (1982) examined when they recruited a sample of adults over a week to participate in unpaid work experience. Findings indicated that perceived voluntariness is associated with positive affect and involvement. It is understandable that lower effect is associated with obligatory work due to the constraints of obligatory work. Self-awareness was found to be associated with lower effect, activation, and personal involvement, but only when the activity is felt to be voluntary and not obligatory. Flow interferes with self-awareness by drawing attention away from the immediate activity itself and depleting the intensity of interaction. The evidence indicates that the ideal experience is one in which the person is engaged in an activity voluntarily and is not focusing attention on the self (Csikszentmihalyi & Figurski, 1982).

Education and Flow
Who knew studying and learning could be enjoyable and stimulating? Csikszentmihalyi believes there is a problem with modern day education and that this is due to the mass production model it is delivered through. He states that the in the first four million years of human evolution, children learned by observing and participating with adults. He believes that flow theory is an appropriate goal of education because it encourages children to observe and participate in learning thereby creating a more rewarding and interesting environment (Whalen, 1999). He does however acknowledge that extrinsic and intrinsic motivators must work together in delivery of education. Extrinsic rewards are important because many skills are difficult at first and they help to overcome that initial inertia. They are however already well manipulated in the education system so it is intrinsic motivation that needs to be focused on (Whalen, 1999).

The participating and stimulating environment that Csikszentmihalyi envisions for the education system already exists in a very similar format known as the Montessori Method of Education. In 2000, Csikszentmihalyi and another psychologist named Kevin Rathunde begun a multi-year study of the Montessori system in comparison to traditional educational systems. As expected, the research found that children in the Montessori method were more likely to experience flow (Rathmunde & Csikszentmihalyi, 2005). Studies have also been conducted in mass production type model schools in order to identify needed changes.

Shernoff, Csikszentmihalyi et al. (2003) examined a sample of high school students in the U.S. to investigate in what conditions students were engaged. In line with flow theory, students experienced increased engagement when perceived challenge of the task and their own skills were both high and in balance. They also experienced engagement when instruction was relevant and the learning environment was under their control. Perceived control and relevance of the activity were also noted as important contributors to engagement. Students spent the majority of studying, in lectures or independent work with only 14% of time being spent in more interactive tasks such as class discussions and activities. Student disengagement may arise from lack of challenge or meaning which was typically found during lectures. Shernoff, Csikszentmihalyi & et.al (2003) argue how students can be expected to reach adult goals of participation and belongingness when active and meaningful participation is not consistently provided in class.

The education system and the concept of Flow must also be incorporated in a manner that is suitable for the age group it is targeted. This is demonstrated in the study by Schweinle, Meyer & Turner (2006) who aimed to explore the relationship between motivation and affect in elementary mathematics classes. Although the study found support for optimal challenge it also found effects contrary to Flow theory. Students in the sample viewed challenge as a threat to their efficacy wherein Flow theory implies challenge should actually increase efficacy (Csikszentmihalyi, Rathunde & Whalen, 1993). The Students in this sample were rather young however and may not have experienced the opportunities offered by optimal challenges. They may also hold a more negative perception of the term 'challenge' than do older students. Further for this age group, challenge may be related more to evaluative situations rather than the desire to improve in an area of interest (Schweinle, Meyer & Turner, 2006).These findings suggest that when developing educational methods, the age and maturity of the students must be considered.

Another consideration that must be taken into account is that students are individuals and therefore require individualized approaches to studying and the teaching they receive. Wong & Csikszentmihalyi (1991) decided to examine this relationship between personality and studying. It was observed that students high in trait intrinsic motivation are more likely to choose challenging and difficult courses. This makes sense because advanced courses are optional and more difficult to handle, so therefore experiential rewards become more important. These students are also probably welcome new challenge, persist in the face of obstacles and maintain their interest in spite of failure. In comparison, students with a high work orientation are more likely to be less self-conscious while studying and would probably study more yet they not need necessarily feel happy, motivated or satisfied about their performance while they study. Therefore they tend to study to achieve long term goals such as good grades (Wong & Csikszentmihalyi, 1991). It is these students that may need extra help in identifying ways to enhance intrinsic motivation.

Applications of Flow in Education
There are three domains in which to improve intrinsic motivation and flow within a student. The first is the student their self. Csikszentmihalyi developed a concept he called overlearning, in which a student should practice newly acquired skills well beyond their point of initial mastery, leading to automaticity. Overlearning enables the mind to concentrate on the desired performance as a singular, integrated action which will increase the ability of flow (Csikszentmihalyi, 1997).

The second and third domains is the students' support structure, that is, their parents and teachers. Parents should provide two things. The first is emotional support, acceptance, and rituals that brings the family together and allows the child to feel that their goals are supported. The second is challenge with high expectations whilst allowing opportunities for developing individuality and privacy. If a parent is committed to a skill for example, learning the piano, and the child is uninterested then Csikszentmihalyi believes the parent is entitled to try if they still support and allow other interests in their life (Whalen, 1999). This is based on the reasoning that some tasks require an external push in order for intrinsic motivation to follow. In regards to education, teachers are perhaps a child's biggest influencer as they are the ones whose approach to teaching can really make a difference.

Schweinle, Meyer & Turner (2006) demonstrated that when feedback from teachers was frequent, elaborative, positive and used to develop understanding, students reported high affect, efficacy and importance. Teachers may be able to enhance engagement by supporting students sense of competency and autonomy such as providing tasks that offer choice (Shernoff, Csikszentmihalyi & et.al, 2003). Another challenge for teachers is to provide tasks slightly too difficult for one's skill level but that can be mastered with the acquisition of new skills. Optimal levels of challenge coupled with positive affect and motivational support can provide contexts most supportive for students enjoyment, efficacy and value (Schweinle, Meyer & Turner, 2006).

Leisure activities & Flow
Does Flow have other applications?

Games
The principles of flow can even be applied to video games. Game designers can benefit from the integration of flow principles into game design. Online gaming is particularly good at creating an environment to achieve flow (Chen, 2007). This is due to the continuous scoring, promotion, immediate feedback and achievement of self-satisfaction that have been recognized as avenues to achieve flow. In a study conducted by Chin-Sheng & Wen-Bin (2006) motivation was explored in regards to online games addiction among adolescents. Results indicated that flow state was negatively correlated with addictive inclination. The flow state of addicts was even lower than that of non-addicts. Need gratification was also examined which found that for addicted players, the absence of playing games was more likely to produce dissatisfaction. That is, the addicts' use of compulsive use of games appears to stem from the relief of dissatisfaction rather than for the pursuit of satisfaction. In comparison, for non-addicts use of online games brought a sense of satisfaction. Why addicts did not experience flow needs to be further examined, however the authors suggest that it may be because the addicts are trying to avoid other aspects of life and in this reasonably mellow state to begin with, are unable to achieve flow while playing (Chin-Sheng & Wen-Bin, 2006).

Television
Csikszentmihalyi et al. has done research examining experience during television viewing. ESM was used to report participants' moods and cognitive states at random times over the course of a week (Kubey & Csikszentmihalyi, 1990). Participants were found to report significantly lower moods before a heavy night of television viewing than before a light night. Television use may be used to escape solitude and negative effect. There was also evidence to suggest that immersing oneself in television may reduce the likelihood that one will wish to engage in more active pursuits.

Benefits of Flow
Flowing through the currents of positivity...

Flow is often regarded as a positive psychology subject (Seligman & Csikszentmihalyi, 2001). Concentration, interest, and enjoyment in an activity must be experienced simultaneously for flow to occur (Csikszentmihalyi, 1997). So not only is Flow a positive experience in its own right, it is also created by a force of positive energy.

Flow is an innately positive experience, producing feel-good emotions such as intense enjoyment and satisfaction (Csikszentmihalyi, 1988). Such feelings may be felt in retrospect however, as all concentration is focused on the task during actual engagement (Csikszentmihalyi, 1990). Flow nurtures the psychological needs of competence and autonomy which in turn are associated with significant increases in mood, enjoyment, esteem and intrinsic motivation (Shernoff, Csikszentmihalyi et.al, 2003). And although future research is required, Flow may even possibly nurture the third psychological need of relatedness due to findings that social experiences and the presence of others who are united help to achieve state Flow (Kowal & Fortier, 1999). Finally, Flow is one of the strongest predictors of trait happiness (Csikszentmihalyi, 2003). These findings support the notion that the experience of Flow helps to achieve well-being.

Flow is also a positive force because it allows for optimal performance and skill development (Csikszentmihalyi, 1988). Research has found that experiencing a Flow state is positively correlated with optimal performance (Schweinle, Meyer & Turner, 2006). Flow also enables personal development and attainment of mastery. When a person engages in a Flow state, they must continually increase the difficulty of the task at hand in order to maintain challenge. Attempting these new, more difficult challenges increases one's ability of skill. Emerging from the flow experience, the individual feels more competent and has a greater sense of efficacy. They are also likely to have experienced personal growth. In addition, after Flow is experienced a higher subsequent motivation to perform well is likely produced in the individual (Csikszentmihalyi, Abuhamdeh, & Nakamura, 2005). Flow is therefore one of the most effective methods in encouraging a continual force of positive and intrinsic motivation within an individual. It is for these reasons that the application of Flow has been suggested for many aspects of life, with one of the biggest being within education. Flow engages students in an enjoyable, intrinsically satisfying manner that many other styles of learning fail to achieve (Seligman & Csikszentmihalyi, 2001). Flow can even enhance creativity and encourage boring tasks and jobs to be challenging and stimulating (Csikszentmihalyi, 1997). Flow has many applications, with many of these being yet to be explored. {{Hide in print|

Revision Quiz
{Flow theory predominantly nurtures which psychological need? - relatedness + competence - autonomy - self-efficacy
 * type=""}

{ Which two factors should both be high but in balance? - Difficulty/Skill - Challenge/Concentration - Achievability/Knowledge + Skill/Challenge
 * type=""}

{ The experience of Flow represents the ultimate form of what? + Intrinsic Motivation - Creativity - Challenge - Positive encouragement
 * type=""}

{ Csikszentmihalyi is most interested in using research on Flow to: - Enhance capability of skills - Read other people's minds + Improve the education system - Create better video-games }}
 * type=""}

Chapter Summary

 * The experience of Flow represents the ultimate form of intrinsic motivation.
 * Flow nurtures the psychological needs of autonomy, relatedness and most importantly, competence.
 * Flow is inextricably linked to emotion and Flow will not be experienced if emotions are ones of depression or anxiety.
 * Mihaly Csikszentmihalyi is the founder and largest researcher of the Flow experience.
 * The model of Flow may differ between cultures, possibly due to differing cultural dispositions.
 * Some personality types/traits may be better conditioned to experience Flow.
 * The ideal set of personality traits a person can possess to experience Flow is known as the 'Autotelic Personality'
 * Flow may be experienced in more environments and conditions than others. Work and Study are most commonly observed.
 * Education is a primary resource to encourage Flow conditions. The Montessori Method of Education aims create a positive environment, conductive to Flow.
 * Flow can be observed and applied in less recognized settings such as video-games and television viewing.
 * Flow has many benefits to both an individual and society. These include improved psychological health and well-being, personal growth, efficacy, happiness, skill and mastery to name a few.


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User:Egm6321.f10.team3.Sudheesh/Homeworks/Homework4

= Problem 2 - Verify Exactness of Bessel Equation =

Given

 * Bessel Equation is given by
 * {| style="width:100%" border="0"

$$
 * $${{x}^{2}}{{y}^{''}}+x{{y}^{'}}+\left( {{x}^{2}}-{{\upsilon }^{2}} \right)y=0$$
 * $$\displaystyle (Eq. 2.1)
 * $$\displaystyle (Eq. 2.1)
 * }
 * }

 * Where, $$\displaystyle\upsilon \in \mathbb{R}$$

Find

 * $$\displaystyle\left( 1 \right)$$ Verify Exactness of $$\displaystyle Eq.2.1$$ using 2 methods

 * Method 1

 * Second conditions of Exactness:
 * {| style="width:100%" border="0" align="left"

$$ $$
 * First Relation:$$\displaystyle f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_{y}
 * First Relation:$$\displaystyle f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_{y}
 * $$\displaystyle (Eq.2.2)
 * }
 * }

 * and

 * {| style="width:100%" border="0" align="left"

$$ $$
 * Second relation:$$\displaystyle f_{xp} + pf_{yp} + 2f_{y} = g_{pp}
 * Second relation:$$\displaystyle f_{xp} + pf_{yp} + 2f_{y} = g_{pp}
 * $$\displaystyle (Eq.2.3)
 * }
 * }

 * Method 2

 * Another form of second Exactness condition for a $$\displaystyle N2-ODE$$:

 * {| style="width:100%" border="0" align="left"

f_0 - \frac {df_1}{dx} + \frac {d^2f_2}{dx^2}=0 $$ $$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.2.4)
 * }
 * }
 * $$\displaystyle\left( 2 \right)$$ If $$\displaystyle Eq.2.1$$ not Exact, see whether it can be made Exact using Integrating Factor Method with:

 * {| style="width:100%" border="0"

$$
 * $$h\left( x,y \right)={{x}^{m}}{{y}^{n}}$$
 * $$\displaystyle (Eq. 2.5)
 * $$\displaystyle (Eq. 2.5)
 * }
 * }

 * That is,find $$\displaystyle (m,n) $$ such that Eq 2.1 is exact.

Solution

 * PART 1: Verify Exactness of Bessel Equation

 * The first condition for exactness of $$\displaystyle N2-ODE$$ is that it is in the following form:
 * {| style="width:100%" border="0" align="left"

F(x,y,p) = f(x,y,p)y'' + g(x,y,p)=0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2.6)
 * }
 * }

 * where

 * {| style="width:100%" border="0" align="left"

p = y' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2.7)
 * }
 * }

 * Now comparing Eq.2.1 with the form of Eq.2.6,we can see that:

 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle f(x,y,p)={{x}^{2}}$$
 * $$\displaystyle (Eq.2.8)
 *
$$\displaystyle (Eq.2.8)
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle g(x,y,p)=x{p}+\left( {{x}^{2}}-{{\upsilon }^{2}} \right)y$$
 *
$$\displaystyle (Eq.2.9)
 *
$$\displaystyle (Eq.2.9)
 * }
 * }

 * Hence the first condition for exactness is satisfied.

 * Now verify second condition of Exactness:

 * (a) Using Method 1- Eq.2.2 and Eq.2.3

 * Taking derivative of Eq.2.8 yields:

 * {| style="width:100%" border="0" align="left"

\begin{array}{*{20}l} {f_x = 2x} & {f_{xx}  = 2} & {f_{xy}  = 0} & {f_{xp}  = 0 }  \\ {f_y = 0} & {f_{yy}  = 0} & {f_{yp}  = 0}\\ \end{array} $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * Similarly taking derivative of Eq.2.9 yields:

 * {| style="width:100%" border="0" align="left"

\begin{array}{*{20}l} {g_x = p+2xy} & {g_{xp} = 1} & {{g}_{y}}={{x}^{2}}-{{\upsilon }^{2}} & {g_{yp}  = 0 }  \\ {g_p = x} & {g_{pp}  = 0} \\ \end{array} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

 * Substituting the above values to Eq.2.3 (first relation) and Eq.2.4 (second relation):

 * {| style="width:100%" border="0" align="left"

 * $$2+2p\times 0+{{p}^{2}}\times 0=1+p\times 0-\left( {{x}^{2}}-{{\upsilon }^{2}} \right)$$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$
 * $$\Rightarrow 2=1-\left( {{x}^{2}}-{{\upsilon }^{2}} \right)$$
 *
$$\displaystyle (Eq.2.10)
 *
$$\displaystyle (Eq.2.10)
 * }
 * }

 * So first relation for second exactness condition is not satisfied.

 * {| style="width:100%" border="0" align="left"

$$
 * $$0+p\times 0+2\times 0=0\Rightarrow 0=0$$, second relation is satisfied
 *
$$\displaystyle (Eq.2.11)
 *
$$\displaystyle (Eq.2.11)
 * }
 * }

 * Since the first relation is not satisfied, the Eq.2.1 is not Exact

 * (b) Using Method 2- Eq.2.4

 * Differentiating $$\displaystyle Eq.2.1$$, partially with respect to $$\displaystyle y,{{y}^{'}},{{y}^{''}}$$,

 * {| style="width:100%" border="0" align="left"

 * $${{f}_{0}}=\frac{\partial F}{\partial y}={{x}^{2}}-{{\upsilon }^{2}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $${{f}_{1}}=\frac{\partial F}{\partial {{y}^{'}}}=x$$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $${{f}_{2}}=\frac{\partial F}{\partial {{y}^{''}}}={{x}^{2}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * Now, $$\frac{\partial {{f}_{1}}}{\partial x}=1$$, and $$\frac{\partial {{x}^{2}}}=2$$
 * }
 * }
 * }

 * Substittuting above values in $$\displaystyle Eq.2.4$$ yields,

 * {| style="width:100%" border="0" align="left"

 * $${{x}^{2}}-{{\upsilon }^{2}}-1+2\ne 0\Rightarrow $$not satisfied and hence the Eq.2.1 is not Exact.
 * }
 * }
 * }

 *
PART 2: Make Bessel Equation exact using Integrating Factor Method

 * Multiply $$\displaystyle Eq.2.1$$ by Integrating Factor ($$\displaystyle Eq.2.5$$)$$\Rightarrow $$

 * {| style="width:100%" border="0" align="left"

$$
 * $$\underbrace_{f}{{y}^{''}}+\underbrace{{{x}^{m+1}}{{y}^{n}}{{y}^{'}}+{{x}^{m+2}}{{y}^{n+1}}-{{x}^{m}}{{y}^{n+1}}{{\upsilon }^{2}}}_{g}=0$$
 *
$$\displaystyle (Eq.2.12)
 *
$$\displaystyle (Eq.2.12)
 * }
 * }

 * New $$\displaystyle f$$ and $$\displaystyle g$$ are:

 * {| style="width:100%" border="0" align="left"

$$
 * $$f\left( x,y,p \right)={{x}^{m+2}}{{y}^{n}}$$
 *
$$\displaystyle (Eq.2.13)
 *
$$\displaystyle (Eq.2.13)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$
 * $$g\left( x,y,p \right)={{x}^{m+1}}{{y}^{n}}p+{{x}^{m+2}}{{y}^{n+1}}-{{x}^{m}}{{y}^{n+1}}{{\upsilon }^{2}}$$
 *
$$\displaystyle (Eq.2.14)
 *
$$\displaystyle (Eq.2.14)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

Analyzing the second relation of second exactness condition ($$\displaystyle Eq.2.3 $$):$$\displaystyle f_{xp} + pf_{yp} + 2f_{y} = g_{pp}$$, it can be noted that most of the terms drop out since $$\displaystyle f$$ is independent of $$\displaystyle p$$. Consequently all derivatives of $$\displaystyle f$$ with respect to $$\displaystyle p$$ become zero. Moreover $$\displaystyle {{g}_{pp}}=0$$ and hence $$\displaystyle Eq.2.3 $$ is simplified to,
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle 2{{f}_{y}}=0$$
 *
$$\displaystyle (Eq.2.15)
 *
$$\displaystyle (Eq.2.15)
 * }
 * }

 * Taking derivative of $$\displaystyle (Eq.2.13)$$ with repect to $$\displaystyle y$$ and substituting to$$\displaystyle (Eq.2.15)$$ yields,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle n{{x}^{m+2}}{{y}^{n-1}}=0$$
 * }
 * }
 * }

 * The above equation is true only when,

 * {| style="width:100%" border="0"

$$\displaystyle n=0$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\displaystyle (Eq. 2.16)
 * }
 * }

 * As $$\displaystyle n=0 $$, the function $$\displaystyle f$$ and $$\displaystyle g$$ can be rewritten as:

 * {| style="width:100%" border="0" align="left"

$$
 * $$f\left( x,y,p \right)={{x}^{m+2}}$$
 *
$$\displaystyle (Eq.2.17)
 *
$$\displaystyle (Eq.2.17)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$
 * $$g\left( x,y,p \right)={{x}^{m+1}}p+{{x}^{m+2}}y-{{x}^{m}}y{{\upsilon }^{2}}$$
 *
$$\displaystyle (Eq.2.18)
 *
$$\displaystyle (Eq.2.18)
 * }
 * }

 * Taking derivative of $$\displaystyle Eq.2.18$$ yields,

 * {| style="width:100%" border="0" align="left"

\begin{array}{*{20}l} {{f}_{x}}=\left( m+2 \right){{x}^{m+1}}, & {{f}_{xx}}=\left( m+2 \right)\left( m+1 \right){{x}^{m}}, & {f_{xy} = 0}, & {f_{xp}  = 0 }  \\ {f_y = 0}, & {f_{yy}  = 0}, & {f_{yp}  = 0}\\ \end{array} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

 * Similarly taking derivative of $$\displaystyle Eq.2.17$$ yields,

 * {| style="width:100%" border="0" align="left"

\begin{array}{*{20}l} {{g}_{x}}=\left( m+1 \right){{x}^{m}}p+\left( m+2 \right){{x}^{m+1}}y-m{{x}^{m-1}}y{{\upsilon }^{2}}\\ {{g}_{xp}}=\left( m+1 \right){{x}^{m}}\\ {{g}_{y}}={{x}^{m+2}}-{{x}^{m}}{{\upsilon }^{2}}\\ {g_{yp} = 0} \\ \end{array} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

 * Substituting above differentials in $$\displaystyle Eq.2.17$$ gives,

 * {| style="width:100%" border="0" align="left"

 * $$\left( m+2 \right)\left( m+1 \right){{x}^{m}}+0+0=\left( m+1 \right){{x}^{m}}+0-\left( {{x}^{m+2}}-{{x}^{m}}{{\upsilon }^{2}} \right)$$
 * }
 * }
 * }

 * Simplifying,

 * {| style="width:100%" border="0" align="left"

 * $$\left( m+2 \right)\left( m+1 \right)=m+1+{{\upsilon }^{2}}-{{x}^{2}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$\Rightarrow {{m}^{2}}+2m+\left( {{x}^{2}}-{{\upsilon }^{2}}+1 \right)=0$$
 * }
 * }
 * }

 * $$\displaystyle m$$ is obtained by solving the above quadratic equation:

 * {| style="width:100%" border="0"

$$m=-1\pm \sqrt{\left( {{\upsilon }^{2}}-{{x}^{2}} \right)}$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\displaystyle (Eq. 2.19)
 * }
 * }

 * So the integration factor is obtained by substituting $$\displaystyle Eq. 2.16 $$ and $$\displaystyle Eq. 2.19 $$ in $$\displaystyle Eq. 2.5 $$:

 * {| style="width:100%" border="0"

$$h={{x}^{-1\pm \sqrt{\left( {{\upsilon }^{2}}-{{x}^{2}} \right)}}}$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\displaystyle (Eq. 2.20)
 * }
 * }

 * Thus Bessel Equation can be made exact using the above integrating factor.