To determine The general solution of the higher order differential equation y ( 8 ) + 8 y ( 4 ) + 16 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 y ( 8 ) + 8 y ( 4 ) + 16 y = 0 . Obtain the characteristic equation as follows. r 8 + 8 r 4 + 16 = 0 ( r 4 + 4 ) 2 = 0 r 4 + 4 = 0 ( r 2 ) 2 − ( 2 i ) 2 = 0 On further simplification, ( r 2 + 2 i ) ( r 2 − 2 i ) = 0 ( r 2 + 2 i ) = 0 or ( r 2 − 2 i ) = 0 r 2 = − 2 i or r 2 − 2 i = 0 ( a + b i ) 2 = − 2 i or ( a + b i ) 2 = 2 i a 2 − b 2 + 2 i a b = 0 − 2 i or a 2 − b 2 + 2 i a b = 0 + 2 i Equate the real and complex number as follows. a 2 − b 2 = 0 ⋯ ( 1 ) 2 a b = − 2 ⋯ ( 2 ) } and a 2 − b 2 = 0 ⋯ ( 3 ) 2 a b = 2 ⋯ ( 4 ) } From the equation (2), 2 a b = − 2 a = − 1 b Substitute a = − 1 b in equation (1)

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The general solution of the higher order differential equation y(8)+8y(4)+16y=0.

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