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## Homework Statement

(Can someone please check my work? Bear with me, this is my first time using LaTex on this forum...)

Find the general solution to the first oder ODE

[tex]

y'-y=e^x

[/tex]

by substituting a series

[tex]y= \sum_{n=0}^\infty a_n x^n[/tex] about [tex]x_0=0[/tex], finding the recurrence relation for a_n, and solving to find an expression for the general term [tex]a_n[/tex] in terms of [tex]a_0[/tex]. What is the radius of convergence of the solution?

## Homework Equations

[tex]e^x = \sum_{n=0}^\infty \frac{x^n}{n!}[/tex]

## The Attempt at a Solution

I plugged y into the DE:

[tex]

\sum_{n=1}^\infty a_n n x^{n-1} - \sum_{n=0}^\infty a_n x^n = \sum_{n=0}^\infty \frac{x^n}{n!}

[/tex]

Then I made all series start at n=0:

[tex]

\sum_{n=0}^\infty a_{n+1} (n+1) x^n - \sum_{n=0}^\infty a_n x^n - \sum_{n=0}^\infty \frac{x^n}{n!} = 0

[/tex]

Bringing together like terms:

[tex]

\sum_{n=0}^\infty x^n (a_{n+1} (n+1) - a_n - \frac{1}{n!}) = 0

[/tex]

Set coefficients equal to zero:

[tex]

a_{n+1} (n+1) - a_n - \frac{1}{n!} = 0

[/tex]

Solving for recurrence relation:

[tex]

a_{n+1} = \frac{n!a_n +1}{(n+1)!}

[/tex]

After plugging in n=0,1,2,3,4... the pattern is:

[tex]

a_n = \frac{a_0 + n}{n!}

[/tex]

Therefore, the solution is:

[tex]

y = a_0 + (a_0 + 1)x+(\frac{a_0 + 2}{2})x^2 + (\frac{a_0 + 3}{6})x^3 +...

[/tex]

I was unsure about how to determine the radius of convergence.

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