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Unformatted text preview: Sample for the Midterm 2 for MA527 1. Find the inverse Laplace transform of s 2 +6 ( s 2)( s 2 +2 s +2) . Solution : Decompose the rational function into two: s 2 + 6 ( s 2)( s 2 + 2 s + 2) = A s 2 + Bs + C s 2 + 2 s + 2 . A, B, C are easily computable by clearing fractions and comparing the numerators: s 2 + 6 = A ( s 2 + 2 s + 2) + ( Bs + C )( s 2) First, substitute s = 2. Then we get 10 = 10 A . Comparing the coefficient of s 2 , we get B = 0, and comparing the constants, we get C = 2. Since s 2 + 2 s + 2 = ( s + 1) 2 + 1, we have an sshift, and hence the answer is L 1 ( 1 s 2 ) 2 L 1 ( 1 s 2 + 2 s + 2 ) = e 2 t 2 e t sin t 2. Find the inverse Laplace transform of e 3 s s 2 +4 s +5 . Solution : First, L 1 ( 1 s 2 + 4 s + 5 ) = L 1 ( 1 ( s + 2) 2 + 1 ) = e 2 t sin t. Therefore by the second shift theorem the answer is equal to e 2( t 3) sin( t 3) u ( t 3) 3. Compute u ( t 1) * ( e 2 t u ( t )) and its Laplace transform. Solution : By definition, u ( t 1) * ( e 2 t u ( t )) = Z t u ( τ 1) e 2( t τ ) u ( t τ ) dτ Note that if τ < 1 then the first factor under the integral sign is zero. Hence if t < 1, then τ < 1, so the integral is 0. If t ≥ 1, then still the integrand is 0 for τ < 1 and reduces to e 2( t τ ) for 1 < τ < t . Hence in this case the integral is equal to e 2 t R t 1 e 2 τ dτ = 1 2 (1 e 2 2 t ). Therefore, the final answer is u ( t 1) * ( e 2 t u ( t )) = 1 2 (1 e 2 2 t ) u ( t 1) The Laplace transform is equal to the product of Laplace transforms, i.e. to L ( u ( t 1)) L ( e 2 t ) = e s s ( s +2) . 4. Solve y ( t ) = 2 t 4 R t y ( τ )( t τ )d τ . Solution : This means y = 2 t 4 y * t and taking Laplace transform we get Y = 2 s 2 4 Y · 1 s 2 . Therefore Y = 2 s 2 +4 and y = L 1 ( 2 s 2 +4 ) = sin 2 t . 5. Solve y 00 + 2 y 3 y = 8 e t + δ ( t 1 / 2) y (0) = 3 y (0) = 5. Solution : Taking Laplace transform, one gets ( s 2 + 2 s 3) Y = 8 s + 1 + e s 2 + ( s + 2) y (0) + y (0) i.e. Y = 8 ( s + 1)( s 1)( s + 3) + e s/ 2 ( s 1)( s + 3) + 3 s + 1 ( s 1)( s + 3) ....
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This note was uploaded on 04/03/2012 for the course MA 527 taught by Professor Weitsman during the Spring '08 term at Purdue University.
 Spring '08
 WEITSMAN
 Advanced Math

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