Finding general solution of linear ODE inhomogeneous

[ohspyro89]

Homework Statement

Find the general solution to y''-2y'-24y=50e^6x-14cos(x)-175sin(x)

Homework Equations

I can't figure out how to solve for B,C,D, and E. I'm wondering if I did something wrong.

The Attempt at a Solution

I'm attaching photos, since it'd take forever to type this all out. I'm stuck on this one, and I'm working on another problem at the same time.

 

[cragar]

it should be factored as (m-6)(m+4)
not (m-6)(m-4) ill look at the rest.

 

[Mark44]

Yes, you did something wrong. Your particular solution should be
y_p = Axe^6x + Bsin(x) + Ccos(x)

The general solution will be y = c₁e^6x + c₂e^-4x + y_p

 

[ohspyro89]

*Face-Palm*

Alright, I'll work on it. Thanks guys!

Also, was I right to get rid of one of the +-i for the undetermined coefficients part? So, now I don't have a D and E to worry about, but it covers the cosx-sinx.

 

[Mark44]

You really don't need either of the +/- i pairs. If your roots of the char. equation include +/- i, the solutions will include e^(ix) and e^(-ix), but you can take a linear combination of these and work with cos(x) and sin(x) instead.

 

[ohspyro89]

How's this? I got a constant to be zero, which seems odd to me. Is this the right general solution?

 

[cragar]

A should equal 5

 

[Mark44]

Looks reasonable - you can check it yourself. Is y'' - 2y' - 24y = 50e^6x-14cos(x)-175sin(x) an identity? IOW, if you replace the 3 terms on the left with your solution and its first two derivatives, do you end up with what you have on the right?