Im still having problems with this simple question:
d^2y/dx^2 + 2 dy/dx +5y=0
With initial conditions y=1, x=0 and dy/dx=1
I have gone through and checked and ended up with no solution given the initial conditions.
The Furthest I have got is ending with 1=C+2D. Which suggests there is no solution but im not sure if im still not on the right track. Can anyone confirm if there is a solution that satisfy the conditions.
As I recall, we got to the point where
y = e^-x (a sin2x + b cos2x)
so,
y' = e^-x((2a-b)cos2x - (a+2b)sin2x))
y(0) = 1 ==> b = 1
y'(0) = 1 ==> 2a-b = 1, so a = 1
Thus,
y = e^-x (sin2x + cos2x)
y' = e^-x (cos2x - 3sin2x)
y" = e^-x (sin2x - 7cos2x)
y" + 2y' + 5y = e^-x ((sin2x-7cos2x)+2(cos2x-3sin2x)+5(sin2x+cos2x)) = 0