University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2022
Views:

72

Pages:

2
Author:

Neveah Bonilla

Review Questions: Exam 3 1. Define the new functions of the section: uc (t) and δ(t − c) (Note: the Dirac function should be defined as a certain limit). Define the new operation, (f ∗ g)(t) 2. 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 3. Check your answers to Problem 2 by rewriting f (t) using the step (or Heaviside) function, and use the table to compute the corresponding Laplace transform. 4. 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) 5. Find the inverse Laplace transform: 2s − 1 − 4s + 6 7 (b) (s + 3)3 e−2s (4s + 2) (c) (s − 1)(s + 2) (a) (d) s2 2s2 3s − 1 − 8s + 14 (e) e−2s − e−3s s2 1 +s−6 6. 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 7. 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 (e) y 00 + y = ∞ X δ(t − 2kπ), y(0) = y 0 (0) = 0. Write your answer in piecewise form. k=1 1 8. Short Answer: Z ∞ π ) dt = 2 0 (b) If y 00 + 2y 0 + 3y = 0 and y(0) = 1, y 0 (0) = −1, compute y 00 (0), y 000 (0), and y (4) (0). (a) sin(3t)δ(t − (c) Using your previous result, give the Taylor expansion of the solution to the differential equation using at least 5 terms. (d) If y 0 (t) = δ(t − c), what is y(t)? (e) 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? (f) Use Laplace transforms to solve for F (s), if f (t) + 2 Z t cos(t − x)f (x) dx = e−t 0 (So only solve for the transform of f (t), don’t invert it back). (g) In order for the Laplace transform of f to exist, f must be (h) Can we assume that the solution to: y 00 + p(x)y 0 + q(x)y = u3 (x) is a power series? √ (i) Is x = 0 an ordinary point for the differential equation: y 00 + x y 0 + xy = x2 ? 9. More on Laplace Transforms: (a) Your friend tells you that the solutions to the IVPs: y 00 +2y 0 +y = 0, y(0) = 0, y 0 (0) = 1 y 00 +2y 0 +y = δ(t) y(0) = 0, y 0 (0) = 0 and are exactly the same. Are they really? Explain. (b) Let f (t) = t and g(t) = u2 (t). i. Use the Laplace transform to compute f ∗ g. ii. Verify your answer by computing f ∗ g using the definition. 10. 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 11. 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. 12. Find the radius of convergence for the following series: (a) ∞ √ X nxn (c) n=1 ∞ X (−2)n √ (b) (x + 3)n n+1 n=1 2 ∞ X n! xn n n=1 n

Math 244 2010 Review Questions Exam 3

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