Math 2500: Differential Equations Spring 2016 Practice Exam #4

Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 1 1. Find the Laplace transform of 𝑓(𝑡) using the integral definition of the Laplace transform 𝑓(𝑡) = { 𝑒 −3𝑡 , 0 < 𝑡 < 3 1, 3 < 𝑡 Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 2 2. Determine the Laplace transform. You may use the Laplace transform table. ℒ[𝑡 2 − 𝑡 + 𝑒 −3𝑡 𝑐𝑜𝑠2𝑡] Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 3 3. Solve the IVP (find y(t)) using the method of the Laplace transform. 𝑦 ′′ + 𝑦 = 𝑡 2 + 2; 𝑦(0) = 1; 𝑦 ′ (0) = −1 Hint: You will NOT need to use partial fractions Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 4 4. Solve for Y(s). You do not need to do a partial fraction expansion, but express your answer as a single rational function with series of factors in the denominator and a number of summed (or differenced) terms in the numerator. 𝑦 ′′ − 6𝑦 ′ + 5𝑦 = 𝑡𝑐𝑜𝑠(2𝑡); 𝑦(0) = 2; 𝑦 ′ (0) = −1 Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 5 5. Consider g(t) below. a. Plot g(t), b. Express g(t) using step functions c. Find the Laplace transform 𝑔(𝑡) = { 5 cos(𝑡), 0 ≤ 𝑡 ≤ 2𝜋 𝑡, 2𝜋 < 𝑡 Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 6 6. Find ℒ −1 [ 𝑠(𝑒 −2𝑠 ) 𝑠 2+9 ] Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 7 7. Solve for y(t). 𝑦 ′′ + 𝑦 = 𝑡𝑢(𝑡 − 3); 𝑦(0) = 0; 𝑦 ′ (0) = 1 Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 8 8. Solve for y(t)). 𝑦 ′′ − 𝑦 = sin(𝑡 − 3)𝑢(𝑡 − 3); 𝑦(0) = 0; 𝑦 ′ (0) = 1 Math 2500: Differential Equations Spring 2016 Practice Exam #4 Instructor: Mel Henriksen Name: 9