Chapter 3.5, Problem 62P

Program Plan Intro

Program Description: Purpose of problem is to obtain the particular functionfor the differential equation $\left(x^2-1\right)y^{″}+2x{y}^{\prime }+2y=x^2-1$ if the complementary function is $y_c\left(x\right)=c_{1}x+c_2\left(1+x^2\right)$ .

Summary introduction: Problem will use a variation of the parameter method for the non-homogeneous equation $y^{″}+P\left(x\right)y^{\prime }+Q\left(x\right)y=f\left(x\right)$ which has a complementary function $y_c\left(x\right)=c_1{y}_1\left(x\right)+c_2{y}_2\left(x\right)$ . The Wronskian of two independent solutions $y_1\text{ and }{y}_2$ of the associated homogeneous equation is $W=W\left(y_1,y_2\right)$ and a particular solution of the non-homogeneous equation is $y_p\left(x\right)=-y_1\left(x\right){\displaystyle \int \frac{y_2\left(x\right)f\left(x\right)}{W\left(x\right)}}dx+y_2\left(x\right){\displaystyle \int \frac{y_1\left(x\right)f\left(x\right)}{W\left(x\right)}}dx$ .