Explanation Given: The differential equation is d 2 y d x 2 = 1 x d y d x − 4 x 2 y . Approach: A second-order homogeneous equation in the form a x 2 y ″ ( x ) + b x y ′ ( x ) + c y ( x ) = 0 is called Cauchy - Euler equation, where a ≠ 0 and b and c are real numbers. If the equation has a singular point then it has solution in the form of y = x r . If the roots of equation a r 2 + b r + c = 0 are real such that r = r 1 = r 2 , then general solution is y ( t ) = c 1 e r t + c 2 t e r t If the roots of equation a r 2 + b r + c = 0 are real such that r 1 ≠ r 2 then general solution is y ( t ) = c 1 e r 1 t + c 2 e r 2 t If the roots of equation a r 2 + b r + c = 0 are imaginary in form of r 1 , 2 = α ± β i then general solution is y ( t ) = e a t ( c 1 cos β t + c 2 sin β t ) Calculation: Consider the given problem. d 2 y d x 2 = 1 x d y d x − 4 x 2 y d 2 y d x 2 − 1 x d y d x + 4 x 2 y = 0 … ( 1 ) As x > 0 multiply equation ( 1 ) by x 2 . x 2 d 2 y d x 2 − x d y d x + 4 y = 0 ... ( 2 ) Let, w ( r , x ) = x r and L is the left hand side of the equation. Differentiate w ( r , x ) = x r upto second derivative. d 2 d r 2 { L [ w ] ( x ) } = d 2 d r 2 x r = d d r r x r − 1 = r ( r − 1 ) x r − 2 Substitute r ( r − 1 ) x r − 2 for y ″ r x r − 1 for y ′ and x r for y in equation ( 2 )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.5, Problem 6E

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The general solution of a differential equation by substitution of y=xr, for x>0.

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Chapter 8.5, Problem 5E

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The general solution of a differential equation by substitution of y = x r , for x > 0 .

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The general solution of a differential equation by substitution of y = x r , for x &gt; 0 .

Solution Summary: The author explains the general solution of a differential equation by substitution of y=xr.

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To find: The general solution of a differential equation by substitution of y=xr, for x>0.

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Chapter 8 Solutions

The general solution of a differential equation by substitution of y = x r , for x > 0 .

To find:

The general solution of a differential equation by substitution of y=xr, for x>0.