Applications of Series — Question 4

*Question 4 Solve the initial-value problem $y'=y$, $y(0)=1$ by assuming a power-series solution $y=\sum_{n=0}^{\infty}a_nx^n$. Derive the coefficient recursion and identify the resulting function.

*Question 4 – Solution

$$y'=\sum_{n=1}^{\infty}n a_nx^{n-1}=\sum_{n=0}^{\infty}(n+1)a_{n+1}x^n.$$ The original series is $y=\sum_{n=0}^{\infty}a_nx^n$.

$$(n+1)a_{n+1}=a_n,\qquad a_{n+1}=\frac{a_n}{n+1}.$$ The initial condition gives $a_0=y(0)=1$. Repeated use of the recursion yields $$a_1=1,\quad a_2=\frac1{2!},\quad a_3=\frac1{3!},\quad\text{and in general}\quad a_n=\frac1{n!}.$$

$$y(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}=\boxed{e^x}.$$ The series has infinite radius by the Ratio Test, so the solution is valid for every real $x$.