Explanation Approach: 1. Make the characteristic equation by substituting y ″ = r 2 , y ′ = r . The characteristic equation is, a r 2 + b r + c = 0 2. Find the roots of the characteristic equation. 3. If roots are real. Case-1 If r = r 1 = r 2 , then the general solution is y ( t ) = c 1 e r t + c 2 t e r t Case-2 If r 1 ≠ r 2 , then the general solution is y ( t ) = c 1 e r 1 t + c 2 e r 2 t 4. If roots are imaginary in the form of r 1 , 2 = α ± β i , then the general solution is y ( t ) = e a t ( c 1 cos β t + c 2 sin β t ) 5. Substitute initial condition in the solution to get the value of integration constant. 6. Substitute value of integration constant to get the solution of initial value problem. Calculation: The characteristic equation is, r 2 + 4 r + 20 = 0 The roots of the above equation are, r 2 + 4 r + 20 = 0 ( r 2 + 2 r ( 2 ) + ( 2 ) 2 ) + 16 = 0 ( r + 2 ) 2 − ( 4 i ) 2 = 0 r = − 2 ± 4 i Here, the roots are imaginary. Substitute − 2 for α and 4 for β in general solution. y ( t ) = e − 2 × t ( c 1 cos ( 4 t ) + c 2 sin ( 4 t ) ) y ( t ) = e − 2 t ( c 1 cos ( 4 t ) + c 2 sin ( 4 t ) ) Differentiate the above equation with respect to t . y ′ ( t ) = d d t ( e − 2 t ( c 1 cos ( 4 t ) + c 2 sin ( 4 t ) ) ) y ′ ( t ) = ( − 2 c 1 + 4 c 2 ) e − 2 t cos 4 t + ( − 4 c 1 − 2 c 2 ) e − 2 t sin 4 t Substitute 0 for t and 2 for y ( 0 ) in the above equation

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Differential Equations

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Chapter 4.2, Problem 27E

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The solution of the initial value problem y″+4y′+20y=0 where y(0)=2, y′(0)=0, and graph the solution on an appropriate interval.

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Chapter 4.2, Problem 26E

Chapter 4.2, Problem 28E

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Chapter 4 Solutions

Introductory Differential Equations

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The solution of the initial value problem y ″ + 4 y ′ + 20 y = 0 where y ( 0 ) = 2 , y ′ ( 0 ) = 0 , and graph the solution on an appropriate interval.

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The solution of the initial value problem y″+4y′+20y=0 where y(0)=2, y′(0)=0, and graph the solution on an appropriate interval.

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Chapter 4 Solutions