Question

Verify (7.19) and (7.20).; write the first equation of (7.19) as $x{D}_X=D_Z$, and find ${D^2}_Z$.

Short answer

Hence proved

$$\begin{gathered} x^2\frac{d^{2}y}{d{x}^2}=\frac{d^{2}y}{d{z}^2}-\frac{dy}{dz} \\ {a}_2\frac{d^{2}y}{d{x}^2}+\left(a_1-a_2\right)\frac{dy}{dz}+a_{0}y=f\left(e^z\right) \end{gathered}$$

Step-by-step solution

Given information

It is given that,

$$\begin{gathered} dx=e^z \\ z=lnx \\ \frac{dz}{dx}=\frac{1}{x} \end{gathered}$$

Differential equation

Definition: A differential equation is an equation that relates one or more unknown functions and their derivatives.

Write expression for equation (7.19) & (7.20)

Write the expression for equation (7.19).

$x^2\frac{d^{2}y}{d{x}^2}=\frac{d^{2}y}{d{z}^2}-\frac{dy}{dz}\dots \left(1\right)$

Write the expression for equation.

$a_2\frac{d^{2}y}{d{x}^2}+\left(a_1-a_2\right)\frac{dy}{dz}+a_{0}y=f\left(e^z\right)\dots ...\left(2\right)$

Prove equation (7.19)

Now,

Prove equation (7.20)

Now,

$$\begin{gathered} a_2{x}^2\frac{d^{2}y}{d{x}^2}+a_{1}x\frac{dy}{dx}+a_{0}y=f\left(x\right) \\ {a}_2\left(\frac{d^{2}y}{d{z}^2}-\frac{dy}{dz}\right)+a_1\frac{dy}{dz}+a_{0}y=f\left(e^z\right) \\ {a}_2\frac{d^{2}y}{d{x}^2}+\left(a_1-a_2\right)\frac{dy}{dz}+a_{0}y=f\left(e^z\right) \end{gathered}$$

Therefore, the proof for equation (7.19) and (7.20) is stated above.