Choice of particular solution to this second order,

Kayden Chandler

Kayden Chandler

Answered question

2022-03-17

Choice of particular solution to this second order, inhomogeneous differential equation
y+2y+5y=xex.
And I'm a bit confused what choice to take as my particular solution yp. I see that the solution to the corresponding homogeneous equation is yh(x)=ex(acos(x)+bsin(x)).
yp(x)=Axex,
which led me to this:
Aex(x+2)+2Aex(x+1)+5Axex=xex,
which clearly cannot be solved for A.
So the next things I want to try is:
yp(x)=xex(Acos(x)+Bsin(x))
and then solve for both A,B. But I'm not intuitively clear why I need two unknown constants A,B instead of just one, namely A which failed.

Answer & Explanation

Jarrett Chan

Jarrett Chan

Beginner2022-03-18Added 6 answers

If you try yp(x)=(A+Bx)ex,
2Bex+(A+Bx)ex+2Bex+2(A+Bx)ex+5(A+Bx)ex=xexA=116,B=18.
So, yp(x)=-116(1-2x)ex

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