12-267/Homework Assignment 7 - Revision history

Vsbdthrsh at 02:11, 1 December 2012

2012-12-01T02:11:20Z

← Older revision		Revision as of 22:11, 30 November 2012
Line 20:		Line 20:
	# Find a differential equation satisfied by <math>y(x):=\sum_{n=0}^\infty\begin{pmatrix}2n\\n\end{pmatrix}x^n</math>.		# Find a differential equation satisfied by <math>y(x):=\sum_{n=0}^\infty\begin{pmatrix}2n\\n\end{pmatrix}x^n</math>.
	# Solve that equation to determine <math>y(x)</math> in general, and <math>\sum_{n=0}^\infty\frac{1}{5^n}\begin{pmatrix}2n\\n\end{pmatrix}</math> in particular.		# Solve that equation to determine <math>y(x)</math> in general, and <math>\sum_{n=0}^\infty\frac{1}{5^n}\begin{pmatrix}2n\\n\end{pmatrix}</math> in particular.

			{{Template:12-267:Dror/Students Divider}}

			[http://imgur.com/a/eeTWI#0 Solutions] [[User:Vsbdthrsh\|Vsbdthrsh]]

Drorbn at 15:19, 18 November 2012

2012-11-18T15:19:14Z

← Older revision		Revision as of 11:19, 18 November 2012
Line 1:		Line 1:
	{{12-267/Navigation}}		{{12-267/Navigation}}
	This assignment is due in class on ~~Tuesday~~ November 20. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.		This assignment is due in class on <span style="color: red;">Friday November 23</span>. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

	'''Task 1.''' Find the radius of convergence of the series		'''Task 1.''' Find the radius of convergence of the series

Drorbn at 18:07, 15 November 2012

2012-11-15T18:07:22Z

← Older revision		Revision as of 14:07, 15 November 2012
Line 7:		Line 7:
	# <math>\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}</math> near <math>x=-\frac12</math>.		# <math>\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}</math> near <math>x=-\frac12</math>.

	'''Task 2.''' Solve the equation <math>(y')^2=1-y^2</math>; <math>y(0)=1</math> using power series up to and including the coefficient of <math>x^4</math>. Then compare your result with the Taylor expansion of the exact solution.		'''Task 2.''' Solve the equation <math>(y')^2=1-y^2</math> with <math>y(0)=0</math> and <math>y'(0)>0</math> using power series up to and including the coefficient of <math>x^5</math>. Then compare your result with the Taylor expansion of the exact solution.

	'''Task 3.''' Find the recurrence relation defining the power series solutions of the following equations:		'''Task 3.''' Find the recurrence relation defining the power series solutions of the following equations:

Drorbn at 16:21, 15 November 2012

2012-11-15T16:21:40Z

← Older revision		Revision as of 12:21, 15 November 2012
Line 7:		Line 7:
	# <math>\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}</math> near <math>x=-\frac12</math>.		# <math>\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}</math> near <math>x=-\frac12</math>.

	'''Task 2.''' Solve the equation <math>(y')^2=1-y^2</math>; <math>y(0)=1</math> using power series up to and including the coefficient of <math>x^5</math>. Then compare your result with the Taylor expansion of the exact solution.		'''Task 2.''' Solve the equation <math>(y')^2=1-y^2</math>; <math>y(0)=1</math> using power series up to and including the coefficient of <math>x^4</math>. Then compare your result with the Taylor expansion of the exact solution.

	'''Task 3.''' Find the recurrence relation defining the power series solutions of the following equations:		'''Task 3.''' Find the recurrence relation defining the power series solutions of the following equations:

Drorbn at 23:41, 14 November 2012

2012-11-14T23:41:28Z

← Older revision		Revision as of 19:41, 14 November 2012
Line 1:		Line 1:
	{{12-267/Navigation}}		{{12-267/Navigation}}
	{{In Preparation}}

	This assignment is due in class on Tuesday November 20. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.		This assignment is due in class on Tuesday November 20. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

Line 17:		Line 15:

	'''Task 4.''' In view of the theorem about convergence of power series solutions (Fuchs' theorem), give a lower bound on the radius of convergence of the series solution of the equation <math>(x^2-2x-3)y''+xy'+4y=0</math> near <math>x=0</math>.		'''Task 4.''' In view of the theorem about convergence of power series solutions (Fuchs' theorem), give a lower bound on the radius of convergence of the series solution of the equation <math>(x^2-2x-3)y''+xy'+4y=0</math> near <math>x=0</math>.

			'''Task 5.'''
			# Find a recurrence relation satisfied by <math>a_n:=\begin{pmatrix}2n\\n\end{pmatrix}</math>.
			# Find a differential equation satisfied by <math>y(x):=\sum_{n=0}^\infty\begin{pmatrix}2n\\n\end{pmatrix}x^n</math>.
			# Solve that equation to determine <math>y(x)</math> in general, and <math>\sum_{n=0}^\infty\frac{1}{5^n}\begin{pmatrix}2n\\n\end{pmatrix}</math> in particular.

Drorbn at 23:15, 14 November 2012

2012-11-14T23:15:01Z

New page

{{12-267/Navigation}}
{{In Preparation}}

This assignment is due in class on Tuesday November 20. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

'''Task 1.''' Find the radius of convergence of the series
# <math>\sum_{n=0}^\infty 2^nx^n</math>.
# <math>\sum_{n=0}^\infty\frac{n}{2^n}x^n</math>.
# <math>\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}</math> near <math>x=-\frac12</math>.

'''Task 2.''' Solve the equation <math>(y')^2=1-y^2</math>; <math>y(0)=1</math> using power series up to and including the coefficient of <math>x^5</math>. Then compare your result with the Taylor expansion of the exact solution.

'''Task 3.''' Find the recurrence relation defining the power series solutions of the following equations:
# <math>y''-xy'-y=0</math> with <math>y(0)=1</math> and <math>y'(0)=0</math>.
# <math>y''-xy'-y=0</math> with <math>y(1)=1</math> and <math>y'(1)=0</math>.
# <math>(1-x)y''+y=0</math> with <math>y(0)=0</math> and <math>y'(0)=1</math>.

'''Task 4.''' In view of the theorem about convergence of power series solutions (Fuchs' theorem), give a lower bound on the radius of convergence of the series solution of the equation <math>(x^2-2x-3)y''+xy'+4y=0</math> near <math>x=0</math>.