Chapter 4.7, Problem 23E

To determine

(a)

To prove:

If $t=e^x$ is substituted in $y\left(t\right)$ and $x$ is a new variable in $Y\left(x\right):=y\left(e^x\right)$, the chain rule implies the following relation:

$t\frac{dy}{dx}=\frac{dY}{dx}$

And

$t^2\frac{d^{2}y}{d{t}^2}=\frac{d^{2}Y}{d{x}^2}-\frac{dY}{dx}$

To determine

(b)

To prove:

If $t=e^x$ is substituted in Cauchy-Euler equation, then it becomes constant coefficient equation for $Y\left(x\right)=y\left(e^x\right)$ that is

$a\frac{d^{2}Y}{d{x}^2}+\left(b-a\right)\frac{dY}{dx}+cY\frac{dY}{dx}=f\left(e^x\right)$

To determine

(c)

To express:

The solutions forms $(8)$ and $(9)$ in terms of $t$.