Chapter 8: Q14P (page 436)

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

Expert verified

Hence proved

$x^{2} \frac{d^{2} y}{d x^{2}} = \frac{d^{2} y}{d z^{2}} - \frac{d y}{d z} a_{2} \frac{d^{2} y}{d x^{2}} + (a_{1} - a_{2}) \frac{d y}{d z} + a_{0} y = f (e^{z})$

Step by step solution

Given information

It is given that,

$d x = e^{z} z = l n x \frac{d z}{d x} = \frac{1}{x}$

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).

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

Write the expression for equation.

$a_{2} \frac{d^{2} y}{d x^{2}} + (a_{1} - a_{2}) \frac{d y}{d z} + a_{0} y = f (e^{z}) \dots . . . (2)$

Prove equation (7.19)

Now,

Prove equation (7.20)

Now,

$a_{2} x^{2} \frac{d^{2} y}{d x^{2}} + a_{1} x \frac{d y}{d x} + a_{0} y = f (x) a_{2} (\frac{d^{2} y}{d z^{2}} - \frac{d y}{d z}) + a_{1} \frac{d y}{d z} + a_{0} y = f (e^{z}) a_{2} \frac{d^{2} y}{d x^{2}} + (a_{1} - a_{2}) \frac{d y}{d z} + a_{0} y = f (e^{z})$

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

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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

Step by step solution

Given information

Differential equation

Write expression for equation (7.19) & (7.20)

Prove equation (7.19)

Prove equation (7.20)

One App. One Place for Learning.

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Verify (7.19) and (7.20).; write the first equation of (7.19) as xDX=DZ, and find D2Z.

Short Answer

Step by step solution

Given information

Differential equation

Write expression for equation (7.19) & (7.20)

Prove equation (7.19)

Prove equation (7.20)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Electric Charge Field and Potential

Scientific Method Physics

Force

Fields in Physics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

Verify (7.19) and (7.20).; write the first equation of (7.19) as $x D_{X} = D_{Z}$ , and find ${D^{2}}_{Z}$ .