Second order ODEs - inhomogeneous

scott.b89 · #1 $Old$ October 25th, 2009, 06:46 AM

Hi,

I have a prblem with a question that I'm working on I've managed to get so far but not sure what my next step should be. I wonder if anyone could point me in the right direction?

I've managed this so far:

QUESTION: Find the general solution of:

d^2 y/ d x^2 - 3dy/dx + 2y = 5 + 6e^7x

Find auxiliary equation: m^2 - 3m + 2
=> (m-1) (m-2) = 0
=> m1 = 1, m2 = 2.

Hence complimentary function is:

Y(x) = Ae^x + Be^2x

Not sure where to go from here. I'm okay with normal inhomogeneous equations but basically..... I hate exponentials lol

Any help would be greatly appreciated! :P

Cheers

galactus · #2 $Old$ October 25th, 2009, 07:02 AM

You're correct, the complementary function is $C_{1}e^{x}+C_{2}e^{2x}$

To use Undetermined Coefficients, try $y_{p}=A+Be^{7x}$

and it will work out.

$y''_{p}=49Be^{7x}, \;\ y'_{p}=7Be^{7x}$

Sub into the original DE, equate coefficients, and solve for A and B.

scott.b89 · #3 $Old$ October 25th, 2009, 07:50 AM

Thanks for that, I think that helps a lot

Cheers mate