I got the following question:
Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$
So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + xC_2e^{-5x} $$ And then calculating the derative while i am at it: $$ y'(x) = -5C_1e^{-5x} -5C_2xe^{-5x} $$ After filling in the conditions, the solution should be: $y = xe^{-5x}$. And that is where I get stuck: I do not get that answer.
First i fill in condition y(0) = 0 $$ 0 = C_1*1 + 0 => C_1 = 0 $$ And then y'(0) = 1: $$ 1 = -5C_1*1 + 0 => C_1 = -1/5 $$ Which is, obviously incorrect. Where did I make the mistake? How do I solve this?
Your mistake is in this line: $$ y'(x) = -5C_1e^{-5x} -5C_2xe^{-5x} \color{red}{+C_2 e^{-5x}} $$ by the product rule.