chapter6.page05

# chapter6.page05 - we ﬁnd L t 2 e 2 s = 2 s-1 3 • Step...

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2. USING Mathcad 5 When using Mathcad together with Laplace transform to solve an ODE a n x ( n ) + a n - 1 x ( n - 1) + ··· + a 1 x 0 + a 0 x = f ( t ) we follow these steps, Step One: Apply Laplace to both sides of equation. Using Mathcad to ﬁnd Laplace transform of f ( t ) . Step Two: Using the linear property and d f ( n ) ( s ) = s n b f ( s ) - s n - 1 f (0) - s n - 2 f 0 (0) - ··· - f ( n - 1) (0) . to ﬁnd an algebraic equation for b x ( s ) , Step Three: Solve the equation obtained in Step Two for b x ( s ) and using Mathcad to ﬁnd inverse transform which will be the solution x ( t ) . Example 2.1 . Find general solution to x 00 + 2 x 0 + 3 x = t 2 e t . Solution Since we want to ﬁnd general solution, we set x (0) = a and x 0 (0) = b Step One Apply Laplace transform to both sides of the equa- tion and ﬁnd Laplace transform for t 2 e t . L ( x 00 + 2 x 0 + 3 x )( s ) = L ( t 2 e 2 )( s ) Type, t^2*e^t[Shift][Ctrl][.]laplace,t,simplify
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Unformatted text preview: we ﬁnd L ( t 2 e 2 )( s ) = 2 ( s-1) 3 • Step Two: Using linear property to ﬁnd an equation for b x ( s ) . L ( x 00 + 2 x + 3 x )( s ) = ( b x 00 + 2 b x + 3 b x ( s ) = s 2 b x ( s )-sa-b + 2( s b x ( s )-a ) + 3 b x ( s ) = ( s 2 + 2 s + 3) b x ( s )-sa-b-2 a The equation for b x ( s ) is ( s 2 + 2 s + 3) b x ( s )-sa-b-2 a = 2 ( s-1) 3 Hence, b x ( s ) = sa + b +2 a ( s 2 +2 s +3) + 2 ( s-1) 3 ( s 2 +2 s +3) • Step Three: Using Mathcad to ﬁnd inverse Laplace trans-form and x ( t ) , we enter, s*a+b+a/(s^2+2s+3)+2/(s-1)^3(s^2+2s+3)[Shift][Ctrl][.]laplace,s...
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