Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.9 Introduction to Differential Equations - 7.9 Exercises: 32

Answer

$$y = C{e^{\frac{{{x^3}}}{3} + x}}$$

Work Step by Step

$$\eqalign{ & \frac{{dy}}{{dx}} = y\left( {{x^2} + 1} \right) \cr & {\text{separate}} \cr & \frac{{dy}}{y} = \left( {{x^2} + 1} \right)dx \cr & {\text{integrate both sides}} \cr & \int {\frac{{dy}}{y}} = \int {\left( {{x^2} + 1} \right)} dx \cr & \ln \left| y \right| = \frac{{{x^3}}}{3} + x + c \cr & {\text{solve for }}y \cr & {e^{\ln \left| y \right|}} = {e^{\frac{{{x^3}}}{3} + x + c}} \cr & y = {e^c}{e^{\frac{{{x^3}}}{3} + x}} \cr & {\text{set }}C = {e^c} \cr & y = C{e^{\frac{{{x^3}}}{3} + x}} \cr} $$
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