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: 18

Answer

$$y\left( t \right) = - 2{e^{ - 4t}} + t + 7$$

Work Step by Step

$$\eqalign{ & \frac{{dy}}{{dt}} = 8{e^{ - 4t}} + 1 \cr & {\text{separate the variables}} \cr & dy = \left( {8{e^{ - 4t}} + 1} \right)dt \cr & {\text{integrate both sides }} \cr & \int {dy} = \int {\left( {8{e^{ - 4t}} + 1} \right)dt} \cr & y\left( t \right) = \frac{8}{{ - 4}}{e^{ - 4t}} + t + C \cr & y\left( t \right) = - 2{e^{ - 4t}} + t + C \cr & {\text{the initial condition }}y\left( 0 \right) = 5{\text{ implies that}} \cr & 5 = - 2{e^{ - 4\left( 0 \right)}} + 0 + C \cr & C = 7 \cr & {\text{so the solution of the initial value problem is}} \cr & y\left( t \right) = - 2{e^{ - 4t}} + t + 7 \cr} $$
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