Hey guys i need help with this problem:
y″−4y′+5y=6e2xsinx
I get the roots from the characteristic equation. r2−4r+5=0 ⟹r=2±i
Then i set up:
yh=c1e2xsinx+c2e2xcosx
And i know that:
yp=e2x(Asinx+Bcosx)x
But the problem arrives when i differentiate. The problem is the x on the end i dont know how to work with it, i can solve the other differential equation types. For this problem i end up with:
−xBcosx−2Bcosx+4Acosx−2xBsinx−xAsinx+2Bsinx
After differentiating and putting it back into the equation. But i dont know what to do with the x's.
Answer
If I may suggest.
You can make your life much easier if you start from the very beginning using y=e2xz⟹y′=e2x(z′+2z)⟹y″=e2x(z″+4z′+4z) This makes the differential equation to be z″+z=6sin(x) and then, almost as you wrote, the particular solution will be zp=x(Asin(x)+Bcos(x)) z′p=sin(x)(A−Bx)+cos(x)(Ax+B) z″p=cos(x)(2A−Bx)−sin(x)(Ax+2B) z″p+zp=2Acos(x)−2Bsin(x)=6sin(x) which is easy to solve.
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