Differential Equations Review questions, Exam 3 2019

Review questions, Exam 3 1. What is the ansatz we use for y in: Chapter 6? Section 5.2? 2. Finish the definitions: • The Heaviside function, uc (t): • The Dirac δ−function: δ(t − c) (Note: the Dirac function should be defined as a certain limit) • Define the convolution: (f ∗ g)(t) • A function is of exponential order if: 3. Use the definition of the Laplace transform to determine L(f ): (a) (b)  f (t) = 3, 6 − t,  0≤t<2 t≥2 f (t) = e−t , 0 ≤ t < 5 −1, t ≥ 5 4. Check your answers to Problem 3 by rewriting f (t) using the step (or Heaviside) function, and use the table to compute the corresponding Laplace transform. 5. Show that f (t) = t3 is of exponential order. Repeat with f (t) = cos(t). (HINT: If needed, you may assume that ln(t) < t for t > 0). 6. Write the following functions in piecewise form (thus removing the Heaviside function): (a) (t + 2)u2 (t) + sin(t)u3 (t) − (t + 2)u4 (t) (b) 4 X unπ (t) sin(t − nπ) n=1 7. Determine the Laplace transform, using the table: (a) t2 e−9t (d) e3t sin(4t) (b) e2t − t3 − sin(5t) (e) et δ(t − 3) (c) t2 y 0 (t) (in terms of Y (s)) (f) t2 u4 (t) 8. Find the inverse Laplace transform, using the table: 3s − 1 − 8s + 14
−2s (e) e − e−3s 2s − 1 − 4s + 6 7 (b) (s + 3)3 e−2s (4s + 2) (c) (s − 1)(s + 2) (a) (d) s2 2s2 1 s2 + s − 6 9. For the following differential equations, solve for Y (s) (the Laplace transform of the solution, y(t)). Do not invert the transform. (a) y 00 + 2y 0 + 2y = t2 + 4t, y(0) = 0, y 0 (0) = −1 (b) y 00 + 9y = 10e2t , y(0) = −1, y 0 (0) = 5 (c) y 00 − 4y 0 + 4y = t2 et , y(0) = 0, y 0 (0) = 0 1 10. Solve the given initial value problems using Laplace transforms: (a) 2y 00 + y 0 + 2y = δ(t − 5), zero initial conditions. (b) y 00 + 6y 0 + 9y = 0, y(0) = −3, y 0 (0) = 10 (c) y 00 − 2y 0 − 3y = u1 (t), y(0) = 0, y 0 (0) = −1 (d) y 00 + 4y = δ(t − π2 ), y(0) = 0, y 0 (0) = 1 ∞ X (e) y 00 + y = δ(t − 2kπ), y(0) = y 0 (0) = 0. Write your answer in piecewise form. k=1 11. For the following, use Laplace transforms to solve, and leave your answer in the form of a convolution: (a) 4y 00 + 4y 0 + 17y = g(t) y(0) = 0, y 0 (0) = 0 (b) y 00 + y 0 + 45 y = 1 − uπ (t), with y(0) = 1 and y 0 (0) = −1. 12. Short Answer: Z ∞ π sin(3t)δ(t − ) dt = (a) 2 0 (b) Use Laplace transforms to solve the first order DE, thus finding which function has the Dirac function as its derivative: y 0 (t) = δ(t − c), y(0) = 0 (c) Use Laplace transforms to solve for F (s), if Z t f (t) + 2 cos(t − x)f (x) dx = e−t 0 (So only solve for the transform of f (t), don’t invert it back). (d) In order for the Laplace transform of f to exist, f must be (e) Can we assume that the solution to: y 00 + p(x)y 0 + q(x)y = u3 (x) is a power series? (f) Is x = 0 an ordinary point for the differential equation: xy 00 + 3x2 y 0 + y = 4? 13. Let f (t) = t and g(t) = u2 (t). (a) Use the Laplace transform to compute f ∗ g. (b) Verify your answer by computing f ∗ g using the definition of the convolution. 14. If a0 = 1, determine the coefficients an so that ∞ X nan xn−1 + 2 n=1 ∞ X an xn = 0 n=0 P∞ an xn . P∞ 15. Write the following as a single sum in the form k=2 ck (x − 1)k (with perhaps a few terms in the front): ∞ ∞ X X n(n − 1)an (x − 1)n−2 + x(x − 2) nan (x − 1)n−1 Try to identify the series represented by n=0 n=1 n=1 16. Characterize ALL (continuous or not) solutions to y 00 + 4y = u1 (t), y(0) = 1, y 0 (0) = 1 (Hint: We could have solved this IVP without Laplace transforms. How?) 2 17. Use the table to find an expression for L(ty 0 ). Use this to convert the following DE into a linear first order DE in Y (s) (do not solve): y 00 + 3ty 0 − 6y = 1, y(0) = 0, y 0 (0) = 0 18. Find the recurrence relation between the coefficients for the power series solutions to the following: (a) 2y 00 + xy 0 + 3y = 0, x0 = 0. (b) (1 − x)y 00 + xy 0 − y = 0, x0 = 0 (c) y 00 − xy 0 − y = 0, x0 = 1 19. Exercises with the table: (a) Use table entries 5 and 14 to prove the formula for 9. (b) Show that you can use table entry 19 to find the Laplace transform of t2 δ(t − 3) (verify your answer using a property of the δ function). (c) Prove (using the definition of L) table entries 12 and 13. (d) Prove (using the definition of L) a formula (similar to 18) for L(y 000 (t)). 20. Find the first 5 terms of the power series solution to ex y 00 + xy = 0 if y(0) = 1 and y 0 (0) = −1. 21. Find the radius of convergence for all of the following, and find the interval of convergence for (b) and (d): (a) (b) ∞ X √ nxn n=1 ∞ X (−2)n √ (x + 3)n n + 1 n=1 3 (c) ∞ X (−1)n n2 (x + 2)n 3n n=1 (d) ∞ X (3x − 2)n n5n n=1