Write down a second order inhomogeneous differential equation whose homogeneous cousin has a solution that is a linear combination of exp(x)sin(2x) and exp(x)cos(2x), while the original equation has RHS:
3xexp(x)sin(2x)+5xexp(x)cos(2x)+7x^3.

What is the shape of the general solution of the inhomogenous differential equation?

Question

Write down a second order inhomogeneous differential equation whose homogeneous cousin has a solution that is a linear combination of exp(x)sin(2x) and exp(x)cos(2x), while the original equation has RHS:
3xexp(x)sin(2x)+5xexp(x)cos(2x)+7x^3.

What is the shape of the general solution of the inhomogenous differential equation?

anonymous · Answer

You want to find an equation such that
$$y_h=C_1e^x\sin2x+C_2e^x\cos2x$$
is the homogeneous solution.

You'll get this sort of solution if the roots to the characteristic equation are $r=1\pm2i$:
$$(r-(1+2i))(r-(1-2i))=0~~\iff~~r^2-2r+5=0$$
which has the differential-equation form,
$$y''-2y'+5y=0$$
You're told that the nonhomogeneous part of the equation is
$$3xe^x\sin2x+5xe^x\cos2x+7x^3$$
so the original DE was
$$y''-2y'+5y=3xe^x\sin2x+5xe^x\cos2x+7x^3$$

anonymous · Answer

oh my gosh you made that look so easy. Is that all there is to it? Maybe I just have a hard time understanding what the question is asking me. Thank you!!!!!

anonymous · Answer

You're welcome!

anonymous · Answer

so to figure out the shape do I just need to try and graph it?