Simon's copy of Power series solution of second order ODE

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Simon Thomas 6 years, 1 month ago

Created this as a copy of Power series solution of second order ODE.

Name	Status	Author	Last Modified
Power series solution of second order ODE	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Simon's copy of Power series solution of second order ODE	draft	Simon Thomas	20/03/2019 12:09

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Question statement

{addSumFunction()}

Seek a power series solution, about $x=0$ , in the form

$y(x)=\sum_{n=0}^{\infty}{a_nx^n},$

of the differential equation

$\simplify{y''+{a1}*x*y'+{b1}*y}=0.$

Take $a_0$ and $a_1$ to be arbitrary constants, and enter the coefficients $a_2$ and $a_3$ as functions of $a_0$ and $a_1$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	number	2
amp2	string	$a_{m+2}$
b1	number	2

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Name

Data type

Value

xxxxxxxxxx
 
random(1..3)*sign(random(-1,1))

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `number`

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Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

In your answer use the symbols a0 and a1 for $a_0$ and $a_1$ respectively. In addition, do not enter decimals.

$a_2=$ Gap 0..

$a_3=$ Gap 1..

Advice

We have

$y(x)=\sum_{n=0}^{\infty}{a_nx^n},$

$y'(x)=\sum_{n=1}^{\infty}{a_nnx^{n-1}},$

and

$y''(x)=\sum_{n=2}^{\infty}{a_nn(n-1)x^{n-2}}.$

Substitute these expressions into the original differential equation to obtain

$\simplify{sum(a_n*n*(n-1)x^(n-2),n=2,infty)+{a1}*sum(a_n*n*x^n,n=1,infty)+{b1}*sum(a_n*x^n,n=0,infty)}=0.$

Now reset the index $m=n-2$ in the first summation, and $m=n$ in the second and third summations to obtain

$\simplify{sum({amp2}*(m+2)*(m+1)x^m,m=0,infty)+{a1}*sum(a_m*m*x^m,m=1,infty)+{b1}*sum(a_m*x^m,m=0,infty)}=0.$

This equation must be valid for all values of $x$ , so the coefficients of like powers of $x$ must vanish. Take $m=0$ to obtain the coefficients of $x^0$ , then

$\simplify{2*a2+{b1}*a0}=0,$

and so

$a_2=\simplify{{-b1}*a0/2}.$

Now take $m=1$ to obtain the coefficients of $x^1$ , so

$\simplify{6*a3+{a1}*a1+{b1}*a1}=0,$

then

$a_3=\simplify{-{a1+b1}*a1/6}.$

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This question is used in the following exam:

Differential Equations (Multiple types) by Simon Thomas in Differential Equations.

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Simon's copy of Power series solution of second order ODE

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Taxonomy: mathcentre

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Simon Thomas 6 years, 1 month ago

Error in variable testing condition

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Generated value: `number`

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Simon's copy of Power series solution of second order ODE

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`