# Answer check: differential initial value problem

solve the IVP

y''-8y'+16y=0

y(0)=2
y'(0)=7

## The Attempt at a Solution

auxiliary equation:
r2-8r+16=0
(r-4)2=0
so we have r=4 with m=2

y(x)= c1e4x+c2xe4x

y(0)--> c1e0 + 0 = 2
c1=2
now we have:
y(x)=2e4x+c2xe4x

take first derivative
y'(x) = 8e4x+c2e4x+4c2xe4x (product rule)

y'(0) ----> 8e0+c2e0+ 0 = 7
8+c2=7
c2=-1

so our final equation is:

y(x)=2e4x-xe4x

Everything seems ok, but just wanted to run it by here to make sure it's all done correctly. thanks all!

Looks fine to me

Mark44
Mentor
It's much easier to check that a potential solution actually works than it is to get the solution. For your problem all you need to do are the following:
1. Verify that y(x) = 2e4x + xe4x satisfies y'' - 8y' + 16y = 0.
2. Verify that y(0) = 2.
3. Verify that y'(0) = 7.

If your solution satisfies the differential equation and initial conditions, you can bask in the warm glow of confidence that you nailed that problem.

......I didn't even think of trying that. Thanks guys!