Explanation Procedure used: General solution to a differential equation: 1. Consider the differential equation, L ( x ) . 2. Find the Auxiliary Equation to homogeneous equation associate to the non-homogeneous differential equation. 3. Find the roots of the DE and determine the complementary solution. Consider the DE x 2 y ″ − ( x 2 + 2 x ) y ′ + ( x + 2 ) y = x 3 and the homogeneous DE associated to x 2 y ″ − ( x 2 + 2 x ) y ′ + ( x + 2 ) y = x 3 is x 2 y ″ − ( x 2 + 2 x ) y ′ + ( x + 2 ) y = 0 . Substitute y 1 = x , y ′ = 1 and y ″ = 0 into the DE x 2 y ″ − ( x 2 + 2 x ) y ′ + ( x + 2 ) y = 0 , x 2 y ″ − ( x 2 + 2 x ) y ′ + ( x + 2 ) y = 0 x 2 ( 0 ) − ( x 2 + 2 x ) ( 1 ) + ( x + 2 ) x = ? 0 − x 2 − 2 x + x 2 + 2 x = ? 0 0 = 0 Thus, the solution y 1 = x is a solution to the homogeneous DE associated to the DE x 2 y ″ − ( x 2 + 2 x ) y ′ + ( x + 2 ) y = x 3 . To show that the method of reduction of order leads to the second solution of the homogeneous equation as well as a particular solution of the nonhomogeneous equation Let y = u x , then y ′ = x u ′ + u and y ″ = x u ″ + 2 u ′

Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's ... Problems, 9th Edition, Single-Term

11th Edition

ISBN: 9781337761000

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4, Problem 38RE

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To verify: The solution y1=x is a solution of homogeneous differential equation associate to the DE x2y″−(x2+2x)y′+(x+2)y=x3. Further show that the method of reduction of the order leads to another solution y2 of the homogeneous equation as well as a particular solution yp of the nonhomogeneous equation. Also, form the general solution of the DE on the interval (0, ∞).

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Chapter 4, Problem 37RE

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Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's ... Problems, 9th Edition, Single-Term

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 - Given that x(t) = c1 cos t + c2 sin t is the...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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Solution Summary: The author explains that the solution of homogeneous differential equation associates to the DE y_1=x.

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