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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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