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2302-practice-mid3-soln

2302-practice-mid3-soln - MAP 2302 Fall 2010 — Midterm 3...

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Unformatted text preview: MAP 2302, Fall 2010 — Midterm 3 Review Problems 1 The exam will cover sections 7.2, 7.3, 7.4, 7.5, 7.6, and 7.8. All topics from this review sheet or from the suggested exercises are fair game. 1 Solve for L { y } given the following initial value problems. a. y ′′- 4 y ′ + 8 y = e 2 t cos 3 t ; y (0) = 1; y ′ (0) = 3. Solution: Let Y ( s ) = L { y } . We have L { y } = Y, L braceleftbig y ′ bracerightbig = sY- y (0) = sY- 1 , L braceleftbig y ′′ bracerightbig = s ( L { y } )- y ′ (0) = s 2 Y- s- 3 , so L braceleftbig y ′′- 4 y ′ + 8 y bracerightbig = ( s 2 Y- s- 3)- 4( sY- 1) + 8 Y = ( s 2- 4 s + 8) Y + (- s + 1) . For the righthand side, we first see that L { cos 3 t } = s s 2 + 9 , so using the rule L braceleftbig e at f ( t ) bracerightbig = F ( s- a ), where F ( s ) = L { f } , we see that L braceleftbig e 2 t cos 3 t bracerightbig = s- 2 ( s- 2) 2 + 9 . Putting these two together and solving for Y , we get Y ( s ) = s − 2 ( s − 2) 2 +9 + s- 1 s 2- 4 s + 8 . b. y ′′ + 2 y ′- 3 y = e t + t + 1; y (0) = 9; y ′ (0) =- 3. Solution: Letting Y ( s ) = L { y } we have L { y } = Y, L braceleftbig y ′ bracerightbig = sY- y (0) = sY- 9 , L braceleftbig y ′′ bracerightbig = s ( L { y } )- y ′ (0) = s 2 Y- 9 s + 3 , MAP 2302, Fall 2010 — Midterm 3 Review Problems 2 so L braceleftbig y ′′ + 2 y ′- 3 y bracerightbig = ( s 2 Y- 9 s + 3) + 2( sY- 9)- 3 Y = ( s 2 + 2 s- 3) Y + (- 9 s- 15) . For the righthand side, we have L braceleftbig e t + t + 1 bracerightbig = 1 s- 1 + 1 s 2 + 1 s . Solving for Y , we get Y = 1 s − 1 + 1 s 2 + 1 s + 9 s + 15 s 2 + 2 s- 3 . c. y ′′- 4 y = braceleftbigg sin t < t < π,- sin t t > π. ; y (0) = y ′ (0) = 0. Solution: Letting Y ( s ) = L { y } we have L { y } = Y, L braceleftbig y ′ bracerightbig = sY- y (0) = sY, L braceleftbig y ′′ bracerightbig = s ( L { y } )- y ′ (0) = s 2 Y, so the lefthand side is transformed to L braceleftbig y ′′- 4 y bracerightbig = s 2 Y- 4 Y = ( s 2- 4) Y. In order to transform the righthand side, we first convert it into Heaviside functions: braceleftbigg sin t < t < π,- sin t t > π....
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2302-practice-mid3-soln - MAP 2302 Fall 2010 — Midterm 3...

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