Explanation Given: The given differential equation is, y ' ' ' − y ' ' + 4 y ' − 4 y = 0 The given functions are, { e x , cos 2 x , sin 2 x } . Approach: Let f 1 , ... , f n be any n functions that are ( n − 1 ) differentiable. The function, W [ f 1 , ... , f n ] = | f 1 ( x ) f 2 ( x ) … f n ( x ) f 1 ' ( x ) f 2 ' ( x ) … f n ' ( x ) ⋮ ⋮ ⋮ f 1 ( n − 1 ) ( x ) f 2 ( n − 1 ) ( x ) … f n ( n − 1 ) ( x ) | is called the Wronskian of f 1 , ... , f n . Let f 1 , ... , f n be n solutions on ( a , b ) of a differential equation, y ( n ) ( x ) + p 1 ( x ) y ( n − 1 ) ( x ) + ... + p n ( x ) y ( x ) = 0 Here, p 1 , p 2 , ... , p n are continuous on ( a , b ) . If at some x 0 in ( a , b ) the solutions satisfy, W [ f 1 , ... , f n ] ( x 0 ) ≠ 0 Then every solution f ( x ) of the differential equation can be expressed as, f ( x ) = c 1 f 1 ( x ) + ... + c n f n ( x ) ... ( 1 ) Here, c 1 , ... c n are constants. The equation ( 1 ) is referred as a general solution of the differential equation. Calculation: The given differential equation is, y ' ' ' − y ' ' + 4 y ' − 4 y = 0 Substitute e x for y in the differential equation. y ' ' ' − y ' ' + 4 y ' − 4 y = e x − e x + 4 e x − 4 e x = 5 e 3 x − 5 e 3 x = 0 … ( 2 ) Substitute cos 2 x for y in the differential equation

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 6.1, Problem 16E

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The general solution of the given differential equation and to verify that the given functions form a fundamental solution set for the given differential equation.

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Chapter 6.1, Problem 15E

Chapter 6.1, Problem 17E

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Fundamentals of Differential Equations and Boundary Value Problems

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The general solution of the given differential equation and to verify that the given functions form a fundamental solution set for the given differential equation.

Solution Summary: The author explains that the given functions form a fundamental solution set for the differential equation, and the general solution is, y=c_1,mathrm...

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To find: The general solution of the given differential equation and to verify that the given functions form a fundamental solution set for the given differential equation.

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To find:

The general solution of the given differential equation and to verify that the given functions form a fundamental solution set for the given differential equation.