Saturday, June 25, 2016

calculus - Differential equation with complex roots and undetermined coefficients


Hey guys i need help with this problem:


y4y+5y=6e2xsinx


I get the roots from the characteristic equation. r24r+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:


xBcosx2Bcosx+4Acosx2xBsinxxAsinx+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=e2xzy=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)) zp=sin(x)(ABx)+cos(x)(Ax+B) zp=cos(x)(2ABx)sin(x)(Ax+2B) zp+zp=2Acos(x)2Bsin(x)=6sin(x) which is easy to solve.


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