Math 244 2012 Review Questions Exam 3

Review Questions: Exam 3 The exam is split about 50/50 between the Laplace transform and Power series solutions. You may not use a calculator for this one. 1. What is the ansatz we use for y in: Chapter 6? Section 5.2-5.3? Section 5.4 (for x2 y 00 + αxy 0 + βy = 0)? 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, 0≤t<2 6 − t, t ≥ 2 f (t) = e−t , 0 ≤ t < 5 −1, t ≥ 5 4. Check your answers to the previous problem by rewriting f (t) using the step (or Heaviside) function, and use the table to compute the corresponding Laplace transform. 5. 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 6. Determine the Laplace transform: (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) 7. Find the inverse Laplace transform: 2s − 1 s2 − 4s + 6 7 (b) (s + 3)3 e−2s (4s + 2) (c) (s − 1)(s + 2) 3s − 1 2s2 − 8s + 14
(e) e−2s − e−3s (a) (d) 1 s2 + s − 6 8. 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 9. 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 1 (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 10. 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. 11. Short Answer: Z ∞ π (a) sin(3t)δ(t − ) dt = 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) What is the expected radius of convergence for the series expansion of f (x) = 1/(x2 + 2x + 5) if the series is based at x0 = 1? (d) 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). (e) In order for the Laplace transform of f to exist, f must be (f) Can we assume that the solution to: y 00 + p(x)y 0 + q(x)y = u3 (x) is a power series? √ (g) Use the table to find the Laplace transform of e−2t sinh( 3 t). (Note: You don’t need the definition the hyperbolic sine to answer this question). (h) Is x = 0 an ordinary point for the differential equation: xy 00 + 3x2 y 0 + y = 4? 12. 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. 13. If a0 = 1, determine the coefficients an so that ∞ X nan xn−1 + 2 n=1 Try to identify the series represented by an xn = 0 n=0 P∞ n=0 14. Write the following as a single sum in the form ∞ X ∞ X an xn . P∞ k k=2 ck (x − 1) n(n − 1)an (x − 1)n−2 + x(x − 2) n=1 (with perhaps a few terms in the front): ∞ X nan (x − 1)n−1 n=1 15. Characterize ALL (continuous or not) solutions to y 00 + 4y = u1 (t), y(0) = 1, y 0 (0) = 1 16. 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 2 17. 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 18. 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)). 19. 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. 20. Determine a lower bound for the radius of convergence of series solutions about each given point x0 for the given differential equation: (x2 − 2x + 5)y 00 + 4xy 0 + y = 0 x0 = 0, x0 = 3 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 (c) ∞ X n! xn (A little tricky) nn n=1 (d) ∞ X (3x − 2)n n5n n=1 22. Exercises from 5.4 and class notes (Euler equation, regular singular points, associated Euler equation) 3