1st Order Differential Equation - Power Series Method

[NicolaiTheDane]

Homework Statement

The Attempt at a Solution

I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution $x = x_{HOM} + x_{Inhom}$, but the $x_{HOM}$ ends up the same, except equal 0, which suggests that the only solution is $c_0 = -t \cdot sin(t)$. as all $c_n=0, n \geq 1, t \in (0,\infty)$. But that can't be right, because that actually constitute a solution?

 

[Celso]

try writing $sin(t)$ as $\sum_{0}^{\infty}\frac{(-1)^n t^{2n+1}}{(2n+1)!}$, simplify the expression and then try to find solutions for which the sum is always going to equal to zero