Explanation Procedure used: General solution of differential equation using variation of parameter. 1 If L is a differential equation a 2 ( x ) y ″ + a 1 ( x ) y ′ + a 0 ( x ) y = g ( x ) . Then convert it into standard from, which is given below. S F = a 2 ( x ) ′ ′ y + a 1 ( x ) y ′ + a 0 ( x ) y = g ( x ) a 2 2 Therefore, y ″ + P ( x ) y ′ + Q ( x ) y = f ( x ) is standard equation. Now determine the complementary solution to the homogeneous equation associated with y ″ + P ( x ) y ′ + Q ( x ) y = f ( x ) . C S : y c = c 1 y 1 ( x ) + c 2 y 2 ( x ) 3 Replace the coefficients c 1 and c 2 with functions u 1 ( x ) and u 2 ( x ) . 4 Determine the differential of functions u 1 ( x ) and u 2 ( x ) by Cramer’s Rule. W is Wronskian. u ′ 1 = W 1 W = − y 2 f ( x ) W and u ′ 2 = W 2 W = y 1 f ( x ) W Here, W = | y 1 y 2 y ′ 1 y ′ 2 | , W 1 = | 0 y 2 f ( x ) y ′ 2 | and W 2 = | y 1 0 y ′ 1 f ( x ) | 5 Integrate the derivatives u ′ 1 and u ′ 2 and obtain the particular solution from y p ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) . 6 Obtain the general solution from y = y c + y p . Calculation: Consider the differential equation y ″ + y = sec 2 x . Compare the differential equation to y ″ + P ( x ) y ′ + Q ( x ) y = f ( x ) . Therefore, f ( x ) = cos 2 x . Homogeneous equation associated to y ″ + y = cos 2 x is y ″ + y = 0 . The auxiliary equation is m 2 + 1 = 0 . Therefore, m 2 + 1 = 0 m = ± i Roots of the auxiliary equation are m 1 = i and m 2 = − i . Hence, the complementary solution is y c = c 1 cos x + c 2 sin x . Consider that y 1 ( x ) = cos x and y 2 ( x ) = sin x . Now compute Wronskian of y 1 ( x ) and y 2 ( x ) . Therefore, W ( y 1 , y 2 ) = | y 1 y 2 y ′ 1 y ′ 2 | = | cos x sin x − sin x cos x | = cos 2 x + sin 2 x = 1 Since W ≠ 0 , the solutions y 1 ( x ) and y 2 ( x ) are linearly independent and forms a set of solutions for the homogeneous equation for y ″ + y = 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The differential equation y ″ + y = sec 2 x by variation of parameters.

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