# Thread: Finding the solution to the initial value problem

1. ## Finding the solution to the initial value problem

The equation is y"+8y'+25y=0 with y(0)= 2 & y'(0)=0

I calculated the general solution and figured out the motion is Underdamped. I can't seem to figure out the solution to the initial value problem.

I found S1 and S2 to be equal -4 +/- 3i ad the Ygen = (k1)^(-4t)cost(t) + i(k2)e^(-4t)sin(t)

3. ## Re: Finding the solution to the initial value problem

Originally Posted by crownvicman
The equation is y"+8y'+25y=0 with y(0)= 2 & y'(0)=0

I calculated the general solution and figured out the motion is Underdamped. I can't seem to figure out the solution to the initial value problem.

I found S1 and S2 to be equal -4 +/- 3i ad the Ygen = (k1)^(-4t)cost(t) + i(k2)e^(-4t)sin(t)
That "(k1)^{-4t}", I presume, is a typo. But you also forgot the "3" in "-4+ 3i". The general solution is $\displaystyle Ygen= k1 e^{-4t}cos(3t)+ k2 e^{-4t}sin(3t)$ (I have absorbed the "i" into k2).

Then $\displaystyle Ygen(0)= k1e^0 cos(0)+ k2 e^0 sin(0)= k1= 2$.

Differentiate Ygen and set that derivative, at x= 0, equal to 0 to find k2.