Question

what's the particular solution for y'''-3y'=9x-8?

Answer 1

The complimentary solution will be found the same way as ever, but with a cubic polynomial to deal with the fact that we are doing a higher order equation\[r^3-3r=r(r^2-3)=0\to r=0,\pm\sqrt3\]so our complimentary is\[y_c=c_1e^{\sqrt3x}+c_2e^{-\sqrt3x}+c_3\]our normal guess for the particular would be\[Y_p=Ax+B\]but we already have a constant in our particular solution, so to keep the solutions linearly independent we will multiply bu x\[Y_p=Ax^2+Bx\]\[Y_p'=2Ax+B\]\[Y_p'''=0\]we put this in our equation and get\[0-3(2Ax+B)=9x-8\]\[A=-\frac32\]\[B=\frac83\]and the sum of the particular and linear solution is then\[y(x)=c_1e^{\sqrt3x}+c_2e^{-\sqrt3x}+c_3-\frac32x^2+\frac83x\]

Answer 2

what's c1 and c2 and c3

Answer 3

unknown constants until we get some initial conditions happening here

Answer 4

but you can't type in those constants

Answer 5

never mind got it!

Answer 6

can you also help me with this one?
y''+25y=5xcos(5x)

Answer 7

I wanna know why that other one I'm working on for you is wrong, but ok... I'll work on this one

Answer 8

yea I'm looking at that one too and thanks!