Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q. 48

Expert-verified

Calculus

Found in: Page 570

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29-52.
$\frac{d y}{d x} = x y + x + y + 1, y (0) = c$

On solving, we get $y (x) = - 1 + (1 + c) e^{\frac{1}{2} (x + 1)^{2}}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

Given the expression $\frac{d y}{d x} = x y + x + y + 1, y (0) = c$

Step 2: Take common and use variable separable method

Calculating, we get

$\frac{d y}{d x} = x (y + 1) + 1 (y + 1) = (x + 1) (y + 1)$

Integrating, we get

$\int \frac{1}{y + 1} d y = \int (x + 1) d x \ln | y + 1 | = \frac{1}{2} (x + 1)^{2} + C y + 1 = e^{\frac{1}{2} (x + 1)^{2} + C} y = - 1 + A e^{\frac{1}{2} (x + 1)^{2}}$

Step 3: Substitute x=0,y=c in the equation and solve

Calculating, we get

$c = - 1 + A A = 1 + c y (x) = - 1 + (1 + c) e^{\frac{1}{2} (x + 1)^{2}}$

Most popular questions for Math Textbooks

a

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29–52.

29. $\frac{d y}{d x} = 3 y, y (0) = 4$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = x^{2} y$

Prove Theorem 6.22 by solving the initial-value problem $\frac{d T}{d t} = k (A - T)$ with T(0) = T0, where k and A are constants.

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

$π \int_{0}^{1} e^{2 x} d x a n d π \int_{0}^{1} (e^{2 x} - 1) d x$

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free