Explanation Given: The equation: t y ' ' ' + ( 1 − t ) y ' ' + t y ' − y = 0 Approach: Substitute y = v ( t ) f ( t ) Calculation: y = f ( t ) v = t v The first, second and third derivatives of y are: y ' = v + t v ' y ' ' = 2 v ' + t v ' ' y ' ' ' = 3 v ' ' + t v ' ' ' Substitute in the given equation to get: t ( 3 v ' ' + t v ' ' ' ) + ( 1 − t ) ( 2 v ' + t v ' ' ) + t ( v + t v ' ) − t v = 0 t 2 v ' ' ' + ( 3 t + t ( 1 − t ) <

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 4.7, Problem 50E

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Reduce given third-order equation to a homogeneous linear second-order equation in the variable.

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Chapter 4.7, Problem 49E

Chapter 4.7, Problem 51E

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Fundamentals of Differential Equations and Boundary Value Problems

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To reduce: Reduce given third-order equation to a homogeneous linear second-order equation in the variable.

Solution Summary: The author explains how to reduce the third-order equation to a homogeneous linear second order equation. The first, second and third derivatives of y are: