undefined Solution The general solution to the differential equation y ″ + 8 y ′ − 9 y = 0 . The general solution to the differential equation y ″ + 8 y ′ − 9 y = 0 is y ( t ) = c 1 e − 9 t + c 2 e t _ . Procedure used: Consider the differential equation a y ″ + b y ′ + c y = 0 . The associated auxiliary equation is, a r 2 + b r + c = 0 . Solving the auxiliary equation gives the roots r 1 and r 2 . (a) If the auxiliary equation has distinct real roots r 1 and r 2 , then y ( t ) = c 1 e r 1 t + c 2 e r 2 t is a general solution. (b) If the auxiliary equation has repeated root r , then y ( t ) = c 1 e r t + c 2 t e r t is a general solution. Calculation: The given differential equation is y ″ + 8 y ′ − 9 y = 0 . The auxiliary equation associated with y ″ + 8 y ′ − 9 y = 0 is r 2 + 8 r − 9 = 0 . Factorize the polynomial r 2 + 8 r − 9 = 0 as follows: r 2 + 8 r − 9 = 0 r 2 + 9 r − r − 9 = 0 r ( r + 9 ) − ( r + 9 ) = 0 ( r − 1 ) ( r + 9 ) = 0 r = 1 and r = − 9 Note that, the roots are distinct and real. Thus, the given differential equation has solution of the form y ( t ) = c 1 e − 9 t + c 2 e t . Therefore, the general solution to the differential equation y ″ + 8 y ′ − 9 y = 0 is y ( t ) = c 1 e − 9 t + c 2 e t _ .

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

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In Problems 1–28, find a general solution to the given differential equation. 1. y ″ + 8 y ′ − 9 y = 0

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