Explanation Procedure used: General solution to differential equation subjected to boundary conditions: 1. Consider the differential equation L and find the roots of auxiliary equation f ( m ) . 2. Find the general solution. 3. Substitute the boundary values to determine the unknown coefficients. 4. Substitute the coefficients into the general solution. Calculation: Consider the differential equation y ″ + 2 y ′ + y = 0 and the auxiliary equation associated to y ″ + 2 y ′ + y = 0 is m 2 + 2 m + 1 = 0 . Calculate the roots of m 2 + 2 m + 1 = 0 with the help of m = ( − b ) ± b 2 − 4 a c 2 ( a ) . m = ( − 2 ) ± ( 2 ) 2 − 4 ( 1 ) ( 1 ) 2 ( 1 ) = − 2 ± 0 2 = − 1 ( twice ) Hence, the roots of the auxiliary equation are m 1 = − 1 and m 2 = − 1 . Both the roots are equal and repeated. Hence, the general solution is, y = c 1 e − x + c 2 x e − x Consider the boundary conditions y ( − 1 ) = 0 and y ′ ( 0 ) = 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4, Problem 40RE

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To solve: The differential equation $y″+2y′+y=0$ subjected to the boundary conditions $y(−1)=0,y′(0)=0$ .

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Chapter 4, Problem 39RE

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The differential equation y ″ + 2 y ′ + y = 0 subjected to the boundary conditions y ( − 1 ) = 0 , y ′ ( 0 ) = 0 .

Solution Summary: The author explains the general solution of the differential equation subjected to boundary conditions.

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To solve: The differential equation $y″+2y′+y=0$ subjected to the boundary conditions $y(−1)=0,y′(0)=0$ .

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