Simple NonHomogenous Differential Equation Question

[shards5]

Homework Statement

y"' -9y" +18y" = 30e^x. Does not matter what the initial conditions are for my question.

Homework Equations

N/A

The Attempt at a Solution

So I factored out the equation and got:
r*(r² - 9r + 18)
Which gives roots r = 0, r = 3, r = 6.
Which gives me the general solution:
y(x) = C₀*1 + C₁e^3x + C₂e^6x
Is the above general solution correct?
Next question I have is whether I solved for the y_p correctly.
y_p = A*e^x
y_p' = A*e^x
y_p" = A*e^x
y_p'" = A*e^x
which by plugging into the equation will give.
A*e^x - 9A*e^x + 18A*e^x = 30e^x
which gives 10A*e^x = 30e^x
which gives A = 3.
Which means my general solution should be.
y(x) = y(x) = C₀*1 + C₁e^3x + C₂e^6x +3*e^x
Is all this correct because I am trying to solve for the "C₀, C₁, etc." and I want to be absolutely sure that the base I am working with is correct and that my error is just a miscalculation with the initial conditions. Thanks.

 

[LCKurtz]

At first glance it looks OK to me. But if you really want to be sure, all you have to do is substitute your general solution back into the DE and see if it works.

 

[shards5]

Hmm didn't think about substituting back into the original DE. Thanks a lot for the tip.