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: 19

Answer

$$y\left( t \right) = {t^2} + 4\ln \left| t \right| + 1$$

Work Step by Step

$$\eqalign{ & y'\left( t \right) = \left( {2{t^2} + 4} \right)/t \cr & y'\left( t \right) = \frac{{2{t^2}}}{t} + \frac{4}{t} \cr & {\text{integrate both sides }} \cr & \int {y'\left( t \right)} = \int {\left( {2t + \frac{4}{t}} \right)dt} \cr & y\left( t \right) = {t^2} + 4\ln \left| t \right| + C \cr & {\text{the initial condition }}y\left( 1 \right) = 2{\text{ implies that}} \cr & 2 = {\left( 1 \right)^2} + 4\ln \left| 1 \right| + C \cr & C = 1 \cr & {\text{so the solution of the initial value problem is}} \cr & y\left( t \right) = {t^2} + 4\ln \left| t \right| + 1 \cr} $$
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