Answer check: differential initial value problem

[bakin]

Homework Statement

solve the IVP

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

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

Homework Equations

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)= c₁e^4x+c₂xe^4x

y(0)--> c₁e⁰ + 0 = 2
c₁=2
now we have:
y(x)=2e^4x+c₂xe^4x

take first derivative
y^'(x) = 8e^4x+c₂e^4x+4c₂xe^4x (product rule)

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

so our final equation is:

y(x)=2e^4x-xe^4x

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

 

[NBAJam100]

Looks fine to me

 

[Mark44]

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) = 2e^4x + xe^4x 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.

 

[bakin]

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