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 with same real roots. y = c 1 e r x + c 2 x e r x (3) Here, r is the root 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. 9 y ″ + 12 y ′ + 4 y = 0 (4) Compare equation (1) and (4). a = 9 b = 12 c = 4 Find the auxiliary equation. Substitute 9 for a , 12 for b and 4 for c in equation (2), ( 9 ) r 2 + ( 12 ) r + ( 4 ) = 0 9 r 2 + 12 r + 3 = 0 ( 3 r + 2 ) 2 = 0 ( 3 r + 2 ) = 0 , ( 3 r + 2 ) = 0 Simplify the equation as follows. 3 r = − 2 , 3 r = − 2 r = − 2 3 , − 2 3 Find the general solution of 9 y ″ + 12 y ′ + 4 y = 0 using equation (3). Substitute − 2 3 for r in equation (3), y = c 1 e − 2 3 x + c 2 x e − 2 3 x (5) Modify equation (5) as follows. y ( x ) = c 1 e − 2 3 x + c 2 x e − 2 3 x (6) Find the value of y ( 0 ) . Substitute 0 for x in equation (6), y ( 0 ) = c 1 e − 2 3 ( 0 ) + c 2 ( 0 ) e − 2 3 ( 0 ) = c 1 e − 0 + 0 = c 1 Substitute 1 for y ( 0 ) , 1 = c 1 Differentiate equation (6) with respect to x

Bundle: Calculus: Early Transcendentals, Loose-Leaf Version, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term

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ISBN: 9781305616691

Author: James Stewart

Publisher: Cengage Learning

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Chapter 17.1, Problem 19E

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To solve: The initial-value problem for differential equation 9y″+12y′+4y=0 , y(0)=1 , y′(0)=0 .

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Chapter 17.1, Problem 18E

Chapter 17.1, Problem 20E

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The initial-value problem for differential equation 9 y ″ + 12 y ′ + 4 y = 0 , y ( 0 ) = 1 , y ′ ( 0 ) = 0 .

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 9y″+12y′+4y=0 , y(0)=1 , y′(0)=0 .

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Chapter 17 Solutions