6.34 Next example

In this case the auxiliary equation is $s^{2}+3s-4=(s+4)(s-1)$, so $\kappa=2$ is not a root. Then the PI is of the form $y={Ce^{2x}.}$ In this case we have $y^{\prime}=2Ce^{2x}$ and $y^{\prime\prime}=4Ce^{2x}$, so that

We now solve for $C$ to obtain $C={\frac{1}{3},}$ so that the particular integral is ${\frac{1}{3}e^{2x}.}$