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 - 1.12 Chapter Review - Additional Problems - Page 110: 17


Separable, $y=\sqrt{2} \times |ln(x)|$

Work Step by Step

Since the equation can be manipulated so that only one variable appears on each side, the differential equation is seperable. To separate variables, multiply each side of the equation by $ydx$ to get $$y(dx)(\frac{dy}{dx}=\frac{2\ln x}{xy})$$ $$ydy=\frac{2\ln x}{x} dx$$ Since each side of the equation is in terms of one variable, you can integrate. $$\int ydy =\int \frac{2\ln x}{x} dx$$ For the second integral, use a u-sub of $u=\ln x$ and $du=dx/x$. $$\frac{1}{2}y^2=2\int udu$$ $$\frac{1}{2}y^2=ln(x)^2$$ Solve for $y$. $$y=\sqrt{2(\ln x)^2}=\sqrt{2} \times |ln(x)|$$
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