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: 2

Answer

$y=-\frac{1}{arctan(x)+C}$

Work Step by Step

Multiply both sides of the equation by $\frac{dx}{y^2}$ to separate variables. $$\frac{dx}{y^2}(\frac{dy}{dx}=\frac{y^2}{x^2+1})$$ $$\frac{dy}{y^2}=\frac{dx}{x^2+1}$$ Since each side is in terms of one variable, you can integrate each side. $$\int \frac{dy}{y^2}=\int \frac{dx}{x^2+1}$$ $$-\frac{1}{y}=arctan(x)+C$$ Solve for $y$. $$y=-\frac{1}{arctan(x)+C}$$
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