6.28 Some PIs and CFs

In Example 6.21, the general solution was $y=\frac{x^{3}}{4}+\frac{c}{x}$. Here $\frac{c}{x}$ is the CF, and is the general solution to the homogeneous equation $\frac{dy}{dx}+\frac{y}{x}=0$. For the PI we can take any particular solution to the (inhomogeneous) equation, e.g. $y=\frac{x^{3}}{4}$.

For Example 6.22, we obtained the general solution

The CF here is $y=c\frac{x+1}{x-1}$. (You may want to check that this is a solution for the homogeneous equation $\frac{dy}{dx}+\frac{2}{x^{2}-1}y=0$.) An obvious choice of PI is $y=\frac{x+1}{x-1}\log|x+1|$.