Explanation Definition used: The Wronskian S represented by W ( { f 1 ( t ) , f 2 ( t ) } ) is given by, W ( S ) = | f 1 ( t ) f 2 ( t ) d d t f 1 ( t ) d d t f 2 ( t ) | = f 1 ( t ) d d t f 2 ( t ) − d d t f 1 ( t ) f 2 ( t ) Here, S = { f 1 ( t ) , f 2 ( t ) } and f 1 ( t ) , f 2 ( t ) are two differentiable functions. If W ( S ) ≠ 0 , then the set of function is linearly independent and linearly dependent if W ( S ) = 0 . Calculation: Consider the given differential equation. y ″ + 6 y ′ + 18 y = 0 (1) The provided set of functions is S = { e − 3 t cos 3 t , e − 3 t sin 3 t } . Assume y 1 = e − 3 t cos 3 t and y 2 = e − 3 t sin 3 t . Differentiate the equation y 1 = e − 3 t cos 3 t with respect to t . y 1 ′ = − 3 e − 3 t cos 3 t − 3 e − 3 t sin 3 t Differentiate the above equation with respect to t . y 1 ′ ′ = ( 9 e − 3 t cos 3 t + 9 e − 3 t sin 3 t ) + ( 9 e − 3 t sin 3 t − 9 e − 3 t cos 3 t ) y 1 ′ ′ = 18 e − 3 t sin 3 t Substitute 18 e − 3 t sin 3 t for y ″ 1 , − 3 e − 3 t cos 3 t − 3 e − 3 t sin 3 t for y ′ 1 and e − 3 t cos 3 t for y 1 in the differential equation y ″ + 6 y ′ + 18 y = 0 . 18 e − 3 t sin 3 t + 6 ( − 3 e − 3 t cos 3 t − 3 e − 3 t sin 3 t ) + 18 ( e − 3 t cos 3 t ) = 0 18 e − 3 t sin 3 t − 18 e − 3 t cos 3 t − 18 e − 3 t sin 3 t + 18 e − 3 t cos 3 t = 0 0 = 0 Thus, the function y 1 = e − 3 t cos 3 t satisfies the given differential equation. Differentiate the equation y 2 = e − 3 t sin 3 t with respect to t . y 2 ′ = − 3 e − 3 t sin 3 t + 3 e − 3 t cos 3 t Differentiate the above equation with respect to t

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ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Differential Equations

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Chapter 4.1, Problem 22E

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To show: The set S={e−3tcos3t,e−3tsin3t} is a fundamental set of solutions for the equation y″+6y′+18y=0.

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Chapter 4.1, Problem 21E

Chapter 4.1, Problem 23E

Chapter 4 Solutions

Introductory Differential Equations

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The set S = { e − 3 t cos 3 t , e − 3 t sin 3 t } is a fundamental set of solutions for the equation y ″ + 6 y ′ + 18 y = 0 .

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To show: The set S={e−3tcos3t,e−3tsin3t} is a fundamental set of solutions for the equation y″+6y′+18y=0.

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Chapter 4 Solutions