undefined Solution To solve: The differential equation d y d x = e x + y y − 1 . The solution of the differential equation d y d x = e x + y y − 1 is − y e − y = e x + C _ . Formula used: u-v integration rule: ∫ u d v = u v − ∫ v d u Calculation: The differential equation is, d y d x = e x + y y − 1 Separate the variables and rewrite the differential equation as follows. d y d x = e x + y y − 1 d y d x = e x y − 1 y − 1 e y d y = e x d x ( y − 1 ) e − y d y = e x d x Integrate on both sides of the equation. ∫ ( y − 1 ) e − y d y = ∫ e x d x Integrate the above integral by the method of u-v substitution. Take u = y − 1 gives d u = d y and d v = e − y d y gives v = − e − y . Thus, the integral becomes, ∫ ( y − 1 ) ︸ u e − y d y ︸ d v = ∫ e x d x ( y − 1 ) ( − e − y ) − ∫ ( − e − y ) d y = e x + C − y e − y + e − y − e − y = e x + C − y e − y = e x + C Thus, the solution of the differential equation d y d x = e x + y y − 1 is − y e − y = e x + C _ .

Student's Solutions Manual for Fundamentals of Differential Equations and Fundamentals of Differential Equations and Boundary Value Problems

9th Edition

ISBN: 9780321977212

Author: Nagle, R. Kent; Saff, Edward B.; Snider, Arthur David

Publisher: PEARSON

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Differential Equations

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Student's Solutions Manual for Fundamentals of Differential Equations and Fundamentals of Differential Equations and Boundary Value Problems

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In Problems 1–30 , solve the equation . 1. d y d x = e x + y y − 1

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