Chapter 4.7, Problem 23E

(a)

To determine

To show: The chain rule implies $t\frac{dy}{dt}=\frac{dY}{dx}\text{ and }{t}^2\frac{d^{2}y}{d{t}^2}=\frac{d^{2}Y}{d{x}^2}-\frac{dY}{dx}$ if the substitution $t=e^x$ is made in the function $y\left(t\right)$ and x is regarded as the new independent variable in $Y\left(x\right)=y\left(e^x\right)$.

(b)

To determine

To show: The substitution $t=e^x$ in the Cauchy-Euler differential equation $a{t}^2{y}^{″}\left(t\right)+bt{y}^{\prime }\left(t\right)+cy=f\left(t\right)$ result in a constant-coefficient equation $a\frac{d^{2}Y}{d{x}^2}+\left(b-a\right)\frac{dY}{dx}+cY=f\left(e^x\right)$ for $Y\left(x\right)=y\left(e^x\right)$.

(c)

To determine

The solution forms for repeated and complex roots in terms of t for the substitution $t=e^x$ in the homogenous form $a{y}^{″}\left(x\right)+b{y}^{\prime }\left(x\right)+cy\left(x\right)=0$.