Chapter 3.5, Problem 57P

You can verify by substitution that $y_c=c_{1}x+c_2{x}^{-1}$ is a complementary function for the nonhomogeneous second-order equation $x^{2}y"+xy\text{'}-y=72x^5$. But before applying the method of’ variation of parameters, you must first divide this equation by its leading coefficient $x^2$ to rewrite it in the standard form $y"+\frac{1}{x}y\text{'}-\frac{1}{x^2}y=72x^3$. Thus $f\left(x\right)=72x^3$ in Eq. (22). Now proceed to solve the equations in (31) and thereby derive the particular solution $y_p=3x^5$ .