1. ## differentiation

Hi, I have this initial value problem: f''(x)+4f'(x)+5f(x)=0 with f(0)=1 and f'(0)=-2

I need to show that f(x)=cos(x)/e^2x is a solution to this initial value problem.

then find the domain of this solution..

Pls help, thanks!

2. Originally Posted by tim_mannire

I need to show that f(x)=cos(x)/e^2x is a solution to this initial value problem.
first step is to find $f'(x)$ and $f''(x)$

3. Originally Posted by tim_mannire
Hi, I have this initial value problem: f''(x)+4f'(x)+5f(x)=0 with f(0)=1 and f'(0)=-2

I need to show that f(x)=cos(x)/e^2x is a solution to this initial value problem.

then find the domain of this solution..

Pls help, thanks!
$f(x) = cosx \times e^{-2x}$

Thus

$f'(x)= (-sinx \times e^{-2x}) + (cosx \times -2e^{-2x})$

$f''(x) = (2e^{-2x} sinx- e^{-2x}cosx) +
(2e^{-2x} sinx + 4e^{-2x} cosx)$

$f''(x) = 4e^{-2x}sinx - 3e^{-2x} cosx$

now you can find compute it for the IVP

4. Originally Posted by harish21

now you can find compute it for the IVP
Substitute what you have found into $f''(x)+4f'(x)+5f(x)$ . Does it equal zero?