Explanation Procedure used: Laplace transformation of first order differential equation is given by, L f ′ ( t ) = L f ( t ) − f ( 0 ) , where f ′ ( t ) is piecewise continuous over close subinterval [ 0 , ∞ ) and L = ∫ 0 ∞ e − s t f ( t ) d t . Method used: The following procedure are used to solve the system of equation, a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 2 x + b 2 y + c 2 z = d 2 Step-1: Write the system of equation in matrix A X = B form, [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] [ x y z ] = [ d 1 d 2 d 3 ] Step-2: Now apply the Cramer rules, x = | d 1 b 1 c 1 d 2 b 2 c 2 d 3 b 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | , y = | a 1 d 1 c 1 a 2 d 2 c 2 a 3 d 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | z = | a 1 b 1 d 1 a 2 b 2 d 2 a 3 b 3 d 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | Calculation: Consider the differential equation, x ′ = 2 x − 3 y + e t y ′ = 4 x − 4 y Apply the Laplace transformation of both side of the system of the equation that yield, s X ( s ) − x ( 0 ) = 2 X ( s ) − 3 Y ( s ) + 1 s − 1 s Y ( s ) − y ( 0 ) = 4 X ( s ) − 4 Y ( s ) Apply the boundary condition, substitute 0 for x ( 0 ) , and 0 for y ( 0 ) in the above system of equation. ( s − 2 ) X ( s ) + 3 Y ( s ) = 1 s − 1 − 4 X ( s ) + ( s + 4 ) Y ( s ) = 0 Convert the system of linear equation in matrix form A X = B , [ s − 2 3 − 4 s + 4 ] [ X ( s ) Y ( s ) ] = [ 1 s − 1 0 ] Apply Cramer rule to solve the system of system, X ( s ) = | 1 s − 1 3 0 s + 4 | | s − 2 3 − 4 s + 4 | , Y ( s ) = | s − 2 1 s − 1 − 4 0 | | s − 2 3 − 4

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ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Chapter 8.6, Problem 7E

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The solution of differential equation x′=2x−3y+et,y′=4x−4y, x(0)=0,y(0)=0 by using Laplace transformation and name other methods that could be used to solve the system of equation.

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Chapter 8.6, Problem 6E

Chapter 8.6, Problem 8E

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Chapter 8 Solutions

Introductory Differential Equations

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The solution of differential equation x ′ = 2 x − 3 y + e t , y ′ = 4 x − 4 y , x ( 0 ) = 0 , y ( 0 ) = 0 by using Laplace transformation and name other methods that could be used to solve the system of equation.

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The solution of differential equation x′=2x−3y+et,y′=4x−4y, x(0)=0,y(0)=0 by using Laplace transformation and name other methods that could be used to solve the system of equation.

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Chapter 8 Solutions