Explanation Given: The differential equation is, d x d t = − x + y d y d t = 2 x The initial values are x ( 0 ) = 0 , y ( 0 ) = 1 . Concept: The Linearity property of Laplace transformation is L [ a f ( t ) + b g ( t ) ] = a L { f ( t ) } + b L { g ( t ) } , where a and b are any constants and f and g be any functions of t . The formula for the transform of a derivative in Laplace transformation is, L { f ( n ) ( t ) } = s n F ( s ) − s n − 1 f ( 0 ) − s n − 2 f ' ( 0 ) − ⋯ − f ( n − 1 ) ( 0 ) , where F ( s ) = L { f ( t ) } . The formula for the inverse Laplace transform of e − a t as below. L − 1 { 1 s + a } = e − a t Calculation: Use the above definition to write the Laplace transform of the given differential equation. L { d x d t } = L { − x + y } L { d y d t } = L { 2 x } Use the linearity property of Laplace transformation. L { d x d t } = − L { x } + L { y } L { d y d t } = 2 L { x } Use formula for Laplace of derivatives, L { x ′ } | t = 0 = s L { x } − f ( 0 ) = s L { x } L { y ′ } | t = 0 = s L { y } − f ( 0 ) = s L { y } − 1 Substitute the Laplace transformation of the derivatives in equation (1). s L { x } = − L { x } + L { y } s L { y } − 1 = 2 L { x } Simplify further and let L { x } = X ( s ) and L { y } = Y ( s ) . ( s + 1 ) X ( s ) − Y ( s ) = 0 − 2 X ( s ) + s Y ( s ) = 1 Solve the above equations by elimination method. X ( s ) = 1 s 2 + s − 2 Y ( s ) = s + 1 s 2 + s − 2 Write X ( s ) as a sum of partial fractions. X ( s ) = 1 s 2 + s − 2 = 1 ( s − 1 ) ( s + 2 ) Then, X ( s ) = A ( s − 1 ) + B ( s + 2 )

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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(b) Let I[y] be a functional of y(x) defined by [[y] = √(x²y' + 2xyy' + 2xy + y²) dr, subject to boundary conditions y(0) = 0, y(1) = 1. State the Euler-Lagrange equation for finding extreme values of I [y] for this prob- lem. Explain why the function y(x) = x is an extremal, and for this function, show that I = 2. Without doing further calculations, give the values of I for the functions y(x) = x² and y(x) = x³.

Please use mathematical induction to prove this

L sin 2x (1+ cos 3x) dx 59

Chapter 4 Solutions

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The system of differential equation with the use of Laplace transforms.

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