To determine The solution of the nonhomogeneous linear ordinary differential equation y ″ + 9 y = sec 3 x . Explanation Procedure used: “Suppose that, the nonhomogeneous linear ordinary differential equation of the form y ″ + p ( x ) y ′ + q ( x ) y = r ( x ) and it is continuous on the interval I . Then, the particular solution y p on I is y p ( x ) = − y 1 ∫ y 2 r W d x + y 2 ∫ y 1 r W d x . Here, y 1 , y 2 form a basis of solutions of the corresponding ordinary differential equation. The value of W is obtained by Wronskian of y 1 , y 2 . That is, W = | y 1 y 2 y ′ 1 y ′ 2 | ” Also, the solution of the ordinary differential equation y ″ + p ( x ) y ′ + q ( x ) y = r ( x ) is the sum of general solution y h and particular solution y p . That is, y ( x ) = y h ( x ) + y p ( x ) . Calculation: Given that, the nonhomogeneous linear ordinary differential equation is y ″ + 9 y = sec 3 x . Compare the equation y ″ + 9 y = sec 3 x with the equation y ″ + p ( x ) y ′ + q ( x ) y = r ( x ) . Here, it is observed that the function p ( x ) = 0 , q ( x ) = 9 and r ( x ) = sec 3 x . First compute the general solution y h of the nonhomogeneous linear ordinary differential equation as follows. The auxiliary equation of y ″ + 9 y = sec 3 x is m 2 + 9 = 0 . m 2 + 9 = 0 m 2 = − 9 m = ± 3 i Thus, the roots of the auxiliary equation m 2 + 9 = 0 are ± 3 i . Here, the roots are complex. So, the general solution y h of the nonhomogeneous linear ordinary differential equation is, y h ( x ) = A cos 3 x + B sin 3 x . Here, it is observed that the function y 1 ( x ) = cos 3 x and the function y 2 ( x ) = sin 3 x . Thus, the value of y ′ 1 ( x ) = − 3 sin 3 x and the value of y ′ 2 ( x ) = 3 cos 3 x . In order to obtain the value of y p , first find the Wronskian value as follows. W = | y 1 y 2 y ′ 1 y ′ 2 | = | cos 3 x sin 3 x − 3 sin 3 x 3 cos 3 x | = 3 cos 2 ( 3 x ) + 3 sin 2 3 x = 3 ( cos 2 ( 3 x ) + sin 2 3 x ) = 3 [ ∵ ( cos 2 ( 3 x ) + sin 2 3 x ) = 1 ] Thus, the value of W is 3

10th Edition

ISBN: 9781119809210

Author: Kreyszig

Publisher: WILEY

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Chapter 2.10, Problem 1P

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The solution of the nonhomogeneous linear ordinary differential equation y″+9y=sec3x.

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Chapter 2.9, Problem 20P

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