# Finding general solution to a differential equation using power series

 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,690 Finding general solution to a differential equation using power series I see more mistakes now that I'm awake. They're easy to see if you just compare the two series. Your solution was $$y_2(x) = a_1\left(x - \frac{x^3}{6} - \frac{x^5}{120} - \frac{x^7}{1680} - \cdots\right)$$ but $$-a_1 e^x = a_1\left(-1 - x - \frac{x^2}{2} - \frac{x^3}{6} - \cdots \right)$$ What I meant about the sign of the "first" term was the sign of the linear term.
 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,690 Why are you insisting on having a closed form expression? Sometimes it's not really possible. Just try writing out more terms and see if you can spot a pattern. It usually doesn't help to simplify as you go because you're looking for a pattern. Anyway, to get only odd powers, you typically write something like $$\sin x = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$$