Explanation Procedure used: 1. Substitute y ″ = r 2 , y ′ = r and y = 1 in the general equation, a y ″ + b y ′ + c y = 0 and obtain the characteristic equation as 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 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 ) . For each function in f i ( t ) in f ( t ) , determine the associate set of functions F i ; 1 ≤ i ≤ m can be obtained as, 5. If f i ( t ) = t m then the associate function is F = { 1 , t , t 2 , ... , t m } . 6. If f i ( t ) = t m e α t then the associate function is F = { e α t , e α t t , e α t t 2 , ... , e α t t m } . 7. If f i ( t ) = t m e α t cos β t or f i ( t ) = t m e α t sin β t then the associate function is, F = { e α t cos β t , e α t t cos β t , e α t t 2 cos β t , ... , e α t t m cos β t , e α t sin β t , e α t t sin β t , e α t t 2 sin β t ,

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 4.3, Problem 13E

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The form of a particular solution for the non-homogeneous equation y″+y=8e2t.

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Chapter 4.3, Problem 12E

Chapter 4.3, Problem 14E

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Chapter 4 Solutions

Introductory Differential Equations

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The form of a particular solution for the non-homogeneous equation y ″ + y = 8 e 2 t .

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Chapter 4 Solutions