To determine The general solution of the higher order differential equation 18 y ‴ + 21 y ″ + 14 y ′ + 4 y = 0 . Explanation Procedure used: General solution of a higher order differential equation. 1. In general, to solve a n th order differential equation a n y n + a n − 1 y n − 1 + ⋯ + a 2 y ″ + a 1 ′ + a 0 y = 0 , it is required to find the solution of a n th order polynomial equation a n r n + a n − 1 r n − 1 + ⋯ + a 0 = 0 . 2. If all the roots of a n r n + a n − 1 r n − 1 + ⋯ + a 0 = 0 are real and distinct, then the general solution will be y = c 1 e r 1 x + c 2 e r 2 x + ... + c n e r n x . 3. If an auxiliary equation has k roots and all the roots of the auxiliary equation are equal, consider that all the roots are equal to r 1 then all the solutions are linearly independent. In such case the general solution of the higher order differential equation must contain the linear combination c 1 e r 1 x + c 2 x e r 1 x + c 3 x 2 e r 1 x + … + c k x k − 1 e r 1 x . 4. In case if higher order differential equation has repeated conjugate complex roots. Then the general solution of the higher order differential equation will be, y = c 1 e r 1 x + c 2 e − r 1 x + c 3 x e r 1 x + c 4 x e − r 1 x where r 1 represents an imaginary root. Calculation: Consider the given differential equation 18 y ‴ + 21 y ″ + 14 y ′ + 4 y = 0 . Obtain the characteristic equation as follows. 18 r 3 + 21 r 2 + 14 r + 4 = 0 ( 2 r + 1 ) ( 9 r 2 + 6 r + 4 ) = 0 2 r + 1 = 0 or ( 9 r 2 + 6 r + 4 ) = 0 r = − 1 2 , 9 r 2 + 6 r + 4 = 0 Find the roots of the equation 9 r 2 + 6 r + 4 = 0 as follows

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 4.2, Problem 25P

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The general solution of the higher order differential equation 18y‴+21y″+14y′+4y=0.

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Chapter 4.2, Problem 24P

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