Chapter 3.5, Problem 52P

Program Plan Intro

Program Description: Purpose of problem is to obtain the particular solution of the differential equation $y^{″}+9y=\sin 3x$ by the method of variation of parameter.

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 $y_p\left(x\right)=-y_1\left(x\right){\displaystyle \int \frac{y_2\left(x\right)f\left(x\right)}{W\left(x\right)}}dx+y_2\left(x\right){\displaystyle \int \frac{y_1\left(x\right)f\left(x\right)}{W\left(x\right)}}dx$ .