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. r^2-4r+5=0 \implies r=2\pm i


Then i set up:


y_{h}=c_{1}e^{2x}sinx+c_{2}e^{2x}cosx


And i know that:


y_p=e^{2x}(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=e^{2x}z \implies y'=e^{2 x} \left(z'+2 z\right)\implies y''=e^{2 x} \left(z''+4 z'+4z\right) This makes the differential equation to be z''+z=6\sin(x) and then, almost as you wrote, the particular solution will be z_p=x(A \sin(x)+B \cos(x)) z'_p=\sin (x) (A-B x)+\cos (x) (A x+B) z''_p=\cos (x) (2 A-B x)-\sin (x) (A x+2 B) z_p''+z_p=2 A \cos (x)-2 B \sin (x)=6\sin(x) which is easy to solve.


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