Chapter 3.1, Problem 51P

(a)

Program Plan Intro

Program Description:Purpose of problem is to show that the substitution of $v=\ln x$ transformation of $a{x}^2{y}^{″}+bx{y}^{\prime }+cy=0$ in to $a\frac{d^{2}y}{d{v}^2}+\left(b-a\right)\frac{dy}{dv}+cy=0$ .

(b)

Program Plan Intro

Program Description:Purpose of problem is to find the general solution of the Euler’s equation.

Summary introduction: Problem will use the characteristic equation of homogeneous second order differential equation $a{y}^{″}+b{y}^{\prime }+cy=0$ is $a{r}^2+br+c=0$ .

If the roots are $r_1\text{ and }{r}_2$ of the characteristic equation are real and distinct, then the general solution of the differential equation is,

  $y\left(x\right)=c_1{e}^{r_{1}x}+c_2{e}^{r_{2}x}$ ....... (3)