If the given is nonhomogeneous differential equation and M and N have different degrees. Is there a solution exist?

Question

Nonhomogeneous Differential Equation

Comments

No differential equation provided, and M and N are not defined.

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M and N are invalid in Math. X and Y are valid in Math.

Answer 1

If M and N are the roots applicable to the homogeneous part of the second order non-homogeneous DE y"+ay'+by=p(x), giving the homogeneous component y1 (characteristic equation) of the solution the form Ae^Mx+Be^Nx, where A and B are constants and where y"+ay'+by=0 (characteristic or auxiliary equation (r-M)(r-N)=r^2+ar+b=0, a=-(M+N) and MN=b), then there will be a solution if a particular solution y2 can be found for p(x). The solution is y1+y2, the sum of the two components.