Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 1 - First-Order Differential Equations - 1.4 Separable Differential Equations - Problems - Page 43: 7

Answer

$y=x-2x^2+3$

Work Step by Step

$$y-x\frac{dy}{dx}=3-2x^2\frac{dy}{dx}$$ $$y-3=x\frac{dy}{dx}-2x^2\frac{dy}{dx}$$ $$y-3=(x-2x^2)\frac{dy}{dx}$$ $$(y-3)dx=(x-2x^2)dy$$ Multiply the entire equation by $\frac{1}{(y-3)(x-2x^2)}$ to separate variables. $$\frac{1}{(y-3)(x-2x^2)}[(y-3)dx=(x-2x^2)dy]$$ $$\frac{dx}{x-2x^2}=\frac{dy}{y-3}$$ Since each side of the equation is in terms of a variable, you can integrate. $$\int \frac{dx}{x-2x^2}=\int \frac{dy}{y-3}$$ $$\ln|x-2x^2|=\ln|y-3|$$ Solve for $y$. $$x-2x^2=y-3$$ $$y=x-2x^2+3$$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.