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Chapter 6: Q. 22 (page 575)

Solve the initial value problem
$\frac{d y}{d x} = y^{2} \sin (x), y (蟺) = 1$

Short Answer

Expert verified

The solution is $y (x) = \frac{1}{2 + \cos (x)}$

Step by step solution

01

Step 1. Given information

Given derivation is $\frac{d y}{d x} = y^{2} \sin (x), y (蟺) = 1$

02

Step 2. Solve the derivation

$鈭� \frac{1}{y^{2}} d y = 鈭� \sin (x) d x - \frac{1}{y} = - \cos (x) + c y = \frac{1}{\cos (x) - c} y (蟺) = 1 1 = \frac{1}{- 1 - c} - 1 - c = 1 c = - 2 s u b s t i t u t e t h e c v a l u e i n e q u a t i o n, t h e s o l u t i o n i s y (x) = \frac{1}{2 + \cos (x)}$

03

Step 3. The solution

The solution is $y (x) = \frac{1}{2 + \cos (x)}$

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Most popular questions from this chapter

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f (x) = 2 {(1 - x)}^{3 / 2} + 3$ , $[a, b] = [- 2, 0]$

$\frac{d y}{d x} = 5 y, y (0) = 1$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals

$蟺 {鈭�}_{0}^{3} x^{2} d x a n d 蟺 {鈭�}_{0}^{3} (9 - x^{2}) d x$

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f (x) = {(2 x + 3)}^{3 / 2}$ , $[a, b] = [- 1, 1]$

How does a slope field help us to understand the solutions of a differential equation? How can a slope field help us sketch an approximate solution to an initial-value problem?

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