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chapter6.page06

# chapter6.page06 - e t L x = L(3 x-4 y t 2 L y = L-x 5 y e t...

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6 1. LAPLACE TRANSFORM METHODS and get x ( t ) = ae - t ± 2 2 sin( 2 t ) + cos( 2 t ) + b 2 2 e - t sin( 2 t ) + e t ± 1 3 t + 2 9 + 2 9 cos( 2 t ) + 2 18 e - t sin( 2 t ) a 2.1. Solving System of equations. We can use Laplace trans- form method to solve system of diﬀerential equations. The procedure is the same as solving a higher order ODE . But, after applying Laplace transform to each equation, we get a system of linear equations whose unknowns are the Laplace transform of the unknown functions. The following example shows how we can use Laplace method with Mathcad to solve system of diﬀerential equations. Example 2.2 . Find solution to x 0 = 3 x - 4 y + t 2 y 0 = - x + 5 y + e t x (0) = 1 , y (0) = - 1 Solution Step One: Apply Laplace transform to both sides of the equa- tions and use Mathcad to ﬁnd Laplace transform of t 2 and
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Unformatted text preview: e t , L ( x ) = L (3 x-4 y + t 2 ) L ( y ) = L (-x + 5 y + e t ) and L ( t 2 )( s ) = 6 s 3 and L ( e t )( s ) = 1 s-1 , • Step Two: Apply linear property to get a system of equations for b x ( s ) and b y ( s ) , due to L ( x ) = s b x ( s )-x (0) and L ( y ) = s b y ( s )-y (0) , s b x ( s )-1 = 3 b x ( s )-4 b y ( s ) + 6 s 3 s b y + 1 =-b x ( s ) + 5 b y ( s ) + 1 s-1 from which we get ( s-3) b x ( s ) + 4 b y ( s ) = 1 + 6 s 3 b x ( s ) + ( s-5) b y =-1 + 1 s-1 Deﬁne b x ( s ) = • b x(s) b y ( s ) ‚ , A = • s-3 4 1 s-5 ‚ , and b ( s ) = • 1 + 6 s 3-1 + 1 s-1 ‚ , we have, in matrix form, A b x ( s ) = b ( s )...
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