Solution of a DE by Laplace Transforms IVP

Solution of a DE by Laplace Transforms IVP - If s=1, C= 2 1...

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Differential Equations Second Quarter SY 0506 ____________________________________________________________________________________________________________ Silva, Alviso and Llacuna – Mathematics Department, MIT 174 Lesson 35 Solution of a Differential Equation by Laplace Transforms (Initial Value Problems) Specific Objectives: At the end of the lesson, the students are expected to: Evaluate differential equations by Laplace transforms The Laplace Operator will transform a linear D.E. with constant coefficients into an algebraic equation in the transformed function (Raindille + Bedient). This method is easily used if the initial conditions are given. Ex. Solve t e y y y 3 2 ' 3 ' ' = + ; 0 ) 0 ( ' ) 0 ( = = y y Find the laplace transforms of both sides. } { } { 2 } ' { 3 } ' ' { 3 t e L y L y L y L = + 3 1 } { 2 )] 0 ( } { [ 3 )] 0 ( ' ) 0 ( [ } { 2 = + + s y L y y sL y sy y L s 3 1 } { 2 ) 0 ( 3 } { 3 0 ) 0 ( } { 2 = + + s y L y sL s y L s 3 1 } { ) 2 3 ( 2 = + s y L s s ) 2 3 )( 3 ( 1 } { 2 + = s s s y L } ) 2 3 )( 3 ( 1 { 2 1 + = s s s L y By Partial Fractions: 1 2 3 ) 1 )( 2 )( 3 ( 1 + + = s C s B s A s s s ) 2 )( 3 ( ) 1 )( 3 ( ) 1 )( 2 ( 1 + + = s s C s s B s s A If s=3, 2 1 = A If s=2, B=-1
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Unformatted text preview: If s=1, C= 2 1 Differential Equations Second Quarter SY 0506 ____________________________________________________________________________________________________________ Silva, Alviso and Llacuna – Mathematics Department, MIT 175 Thus, } 1 2 1 2 1 3 2 1 { 1 − + − − − − = − s s s L y t t t e e e y 2 1 2 1 2 3 + − = Seatwork: Perform the indicated operations: 1. ) 3 sin( 5 ' 2 ' ' t y y y = + + ; 1 ) ( ' , 1 ) ( − = = y y Ans: + + − − + = − − − ) 2 sin( 2 3 ) 2 cos( ) 3 sin( 3 2 ) 3 cos( 26 3 ) 2 cos( ) ( 2 t e t e t t t e t y t t 2. t y y sin 10 3 ' = + ; ) ( = y Ans: t t e t sin 3 cos 3 + − − Homework: Perform the indicated operations: 1. t y y = − ' ' ; 1 ) ( ' , 1 ) ( = = y y 2. t y y y = + + 10 ' 5 ' ' ; 1 ) ( ' ) ( = = y y...
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Solution of a DE by Laplace Transforms IVP - If s=1, C= 2 1...

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