D.E.s particular integrals?
2dy/dx + 4y = 8x + e^-x
I've already found the complimentary function, but I am struggling to find the particular integral. I am a little confused as 8x + e^(-x) is not in the form ce^(kx) or polynomial of degree n, it seems to be a mixture of the two. Help!!!
Thank you very much
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2dy/dx + 4y = 8x + e^-x
=> dy/dx + 2y = 4x + (1/2)e^-x
For DE of the type
dy/dx + Py = Q, where P and Q are functions of x or constant, multiply by e^∫ Pdx
=> e^2x dy/dx + 2ye^2x = 4xe^2x + (1/2) e^x
=> d[ye^2x] = [4xe^2x + (1/2) e^x] dx
Integrating,
ye^2x = 4 ∫ xe^2x dx + (1/2) ∫ e^x dx
=> ye^2x = 4 [x ∫ e^2x dx - ∫ [d/dx(x) ∫ e^2x dx]dx ] + (1/2) e^x
=> y e^2x = 2xe^2x - e^2x + (1/2) e^x + c.
=> y = ce^(-2x) + 2x - 1 + (1/2) e^(-x).