Explanation Formula used: Write the expression for differential equation. a y ″ + b y ′ + c y = 0 (1) Write the expression for auxiliary equation. a r 2 + b r + c = 0 (2) Write the expression for general solution of a y ″ + b y ′ + c y = 0 . y = c 1 e r 1 x + c 2 e r 2 x (3) Here, r 1 and r 2 are the roots of auxiliary equation. Write the required differential formulae to evaluate the differential equation. d d x e n x = n e n x Consider the differential equation as follows. 3 y ″ − 2 y ′ − y = 0 (4) Compare equation (1) and (4). a = 3 b = − 2 c = − 1 Find the auxiliary equation. Substitute 3 for a , − 2 for b and − 1 for c in equation (2), ( 3 ) r 2 + ( − 2 ) r + ( − 1 ) = 0 3 r 2 − 2 r − 1 = 0 ( 3 r + 1 ) ( r − 1 ) = 0 Simplify the equation as follows. 3 r + 1 = 0 r = − 1 3 r − 1 = 0 r = 1 Consider the value of r 1 and r 2 as follows. r 1 = − 1 3 r 2 = 1 Find the general solution of 3 y ″ − 2 y ′ − y = 0 using equation (3). Substitute − 1 3 for r 1 and 1 for r 2 in equation (3), y = c 1 e − 1 3 x + c 2 e x y = c 1 e − x 3 + c 2 e x (5) Modify equation (5) as follows. y ( x ) = c 1 e − x 3 + c 2 e x (6) Find the value of y ( 0 )

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Differential Equations

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Chapter 17.1, Problem 20E

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To solve: The initial-value problem for differential equation 3y″−2y′−y=0 , y(0)=0 , y′(0)=−4 .

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Chapter 17.1, Problem 19E

Chapter 17.1, Problem 21E

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Calculus: Early Transcendentals

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The initial-value problem for differential equation 3 y ″ − 2 y ′ − y = 0 , y ( 0 ) = 0 , y ′ ( 0 ) = − 4 .

Solution Summary: The author explains the formula used to solve the initial-value problem for differential equation.

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To solve: The initial-value problem for differential equation 3y″−2y′−y=0 , y(0)=0 , y′(0)=−4 .

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