Explanation Given information: The given linear differential equation is y ' ' + 2 y ' + 3 y = 0 . The initial condition are y ( 0 ) = 2 , y ' ( 0 ) = 1 Concept Used: If m 1 = α + i β and m 2 = α − i β are complex zeros of the characteristics equation, then the general solution is y = C 1 e α x cos β x + C 2 e α x sin β x . Calculation: The characteristics equation is m 2 + 2 m + 3 = 0 . First, we have to find the solution of the characteristics equation m 2 + 2 m + 3 = 0 . m 1 , 2 = − ( 2 ) ± ( 2 ) 2 − 4 ( 1 ) ( 3 ) 2 ( 1 ) = − 2 ± 2 2 i 2 = − 1 ± 2 i Here, m 1 = − 1 + 2 i and m 2 = − 1 − 2 i are complex zeros of the characteristics equation. Substitute the value of α = − 1 and β = 2 in an equation y = C 1 e α x cos β x + C 2 e α x sin β x , we get y = C 1 e ( − 1 ) x cos ( 2 ) x + C 2 e ( − 1 ) x sin ( 2 ) x = C 1 e − x cos 2 x + C 2 e − x sin 2 x Therefore, the general solution of the given linear differential equation is y = C 1 e − x cos 2 x + C 2 e − x sin 2 x

11th Edition

ISBN: 8220103600781

Author: Edwards

Publisher: YUZU

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Chapter 16.2, Problem 43E

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The particular solution of the linear differential equation y''+2y'+3y=0 that satisfies the initial conditions.

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Chapter 16.2, Problem 42E

Chapter 16.2, Problem 44E

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Chapter 16 Solutions

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The particular solution of the linear differential equation y ' ' + 2 y ' + 3 y = 0 that satisfies the initial conditions.

Solution Summary: The author explains how to calculate the linear differential equation y''+2y satisfies the initial conditions.

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To calculate:

The particular solution of the linear differential equation y''+2y'+3y=0 that satisfies the initial conditions.

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Chapter 16 Solutions

The particular solution of the linear differential equation y ' ' + 2 y ' + 3 y = 0 that satisfies the initial conditions.

Solution Summary: The author explains how to calculate the linear differential equation y''+2y satisfies the initial conditions.

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To calculate: The particular solution of the linear differential equation y''+2y'+3y=0 that satisfies the initial conditions.

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Chapter 16 Solutions

To calculate:

The particular solution of the linear differential equation y''+2y'+3y=0 that satisfies the initial conditions.