Explanation Consider differential equation and function, x y ″ − x y ′ + y = 0 y 1 ( x ) = x If the given solution of differential equation is right then it must be satisfies the differential equation. y 1 ( x ) = x , y 1 ' = 1 , y 1 " = 0 x y 1 " − x y 1 ' + y = 0 x ⋅ 0 − x ⋅ 1 + x = 0 0 = 0 Therefore, it must be solution of the given differential equation. Assume that the solution of given homogenous equation, y ( x ) = u ( x ) y 1 ( x ) y ( x ) = u ( x ) y 1 ( x ) y ( x ) = u ( x ) x y ′ ( x ) = u ( x ) + x u ′ ( x ) y ″ ( x ) = u ′ ( x ) + u ′ ( x ) + x u ″ ( x ) y ″ ( x ) = 2 u ′ ( x ) + x u ″ ( x ) Substitute these values in the given differential equation, x y ″ − x y ′ + y = 0 x ( 2 u ′ ( x ) + x u ″ ( x ) ) − x ( u ( x ) + x u ′ ( x ) ) + x u ( x ) = 0 2 x u ′ ( x ) + u ″ ( x ) x 2 − x u ( x ) − x 2 u ′ ( x ) + x u ( x ) x = 0 u ″ ( x ) x 2 − ( x 2 − 2 x ) u ′ ( x ) = 0 Convert this second order linear equation into the first order linear equation by taking u ′ = t

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.2, Problem 24E

To determine

The second solution y2(x) of the homogenous equation in the form of an infinite series, if the indicated function y1(x)=x is a solution of the differential equation xy″−xy′+y=0 and form an interval of definition for y2(x).

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Chapter 3.2, Problem 23E

Chapter 3.2, Problem 25E

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Chapter 3 Solutions

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The second solution y 2 ( x ) of the homogenous equation in the form of an infinite series, if the indicated function y 1 ( x ) = x is a solution of the differential equation x y ″ − x y ′ + y = 0 and form an interval of definition for y 2 ( x ) .

Solution Summary: The author explains that the second solution of the homogenous equation, y_2(x)=x, must be satisfies the differential equation.

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The second solution y2(x) of the homogenous equation in the form of an infinite series, if the indicated function y1(x)=x is a solution of the differential equation xy″−xy′+y=0 and form an interval of definition for y2(x).

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