6.35 Finding the PI in exceptional cases

Suppose in the previous slide the equation was $y^{\prime\prime}+3y^{\prime}-4y=2e^{x}$ instead of $2e^{2x}$. We can’t have a PI of the form $Ce^{x}$ since this is already a solution of the homogenous equation. In these circumstances we look for a PI of the form $Cxe^{x}$.

This explains the need for the special cases (“$\kappa$ a single root” etc.) on slide 6.31. When $\kappa$ is a double root of the auxiliary equation, not only $e^{\kappa x}$ but also $xe^{\kappa x}$ is a solution to the homogeneous equation and we look for a PI of the form $Cx^{2}e^{\kappa x}$.

The same goes for polynomials: if $1$ is a solution to the homogeneous equation then to find the PI for the case where $q(x)$ is a polynomial of degree $n$, we consider polynomials of degree $n+1$.