Chapter 8: Q4P (page 403)

Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.
$2 x y + y = 2 x^{5 / 2}$

Short Answer

Expert verified

Answer:

The general solution of the differential equation is $y = \frac{x^{5 / 2}}{3} + \frac{C}{X^{1 / 2}}$ .

Step by step solution

Given information

The given differential equation is $2 x y + y = 2 x^{5 / 2}$ .

Meaning of the first-order differential equation

A first-order differential equation is defined by two variables, x and y, and its function $f (x, y)$ is defined on an XY-plane region.

Find the general solution

Write this differential equation to make it in the form $y + P y = Q$ ; that is

$y + \frac{y}{2 x} x^{3 / 2}$ .

From equation 3.4,

$I = \int \frac{d x}{2 x} = \frac{1}{2} I n (x) e' = e^{\frac{1}{2} I n (x)} = x^{1 / 2}$ .

Find a solution for this differential equation

role="math" localid="1655205234482" $y e' = \int (x^{3 / 2}) x^{1 / 2} d x = \frac{x^{3}}{3} + C y = \frac{x^{5 / 2}}{3} + \frac{C}{x^{1 / 2}}$

Therefore, the general solution of the differential equation is $y = \frac{x^{5 / 2}}{3} + \frac{C}{x^{1 / 2}}$ .

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Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.
$2 x y + y = 2 x^{5 / 2}$

Short Answer

Step by step solution

Given information

Meaning of the first-order differential equation

Find the general solution

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Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after (3.9), and Example 1.2xy+y=2x5/2

Short Answer

Step by step solution

Given information

Meaning of the first-order differential equation

Find the general solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Famous Physicists

Space Physics

Solid State Physics

Quantum Physics

Linear Momentum

Study anywhere. Anytime. Across all devices.

Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.
$2 x y + y = 2 x^{5 / 2}$