exam3sampleq

# exam3sampleq - 9 y 00 3 y 2 y = 0 y(0 =-1 y(0 = 3 10 y 00...

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Sample questions for Exam 3 1. Find the general solution to y 00 - 2 y 0 - 8 y = te 4 t + 3 e t using variation of parameters. 2. Find the solution to y 00 + y 0 - 2 y = 5 e 3 t , y (0) = 1 , y 0 (0) = - 1 using (a) variation of parameters and (b) Laplace transforms. 3. Find the general solution to y 00 + 9 y = sec 3 t. 4. Solve the initial value problem y 00 - 6 y 0 + 9 y = 6 e 3 t t, y (0) = 1 , y 0 (0) = 1 . 5. Use variation of parameters to ﬁnd a particular solution to t 2 y 00 + ty 0 - y = 1 . Then verify that your answer is a solution by substituting into the equation. Note: y 1 ( t ) = t and y 2 ( t ) = t - 1 are solutions to the homogeneous equation. 6. Write the function f ( t ) = e - t if 0 t 2 t + 1 if 2 < t 5 6 if 5 < t 6 3 - t if t > 6 in terms of Heaviside functions u c ( t ), and ﬁnd the Laplace transform F ( s ). 7. Graph the function f ( t ) = t - 1 + u 2 ( t )(3 - t ) + 2 u 3 ( t ) t , and ﬁnd the Laplace trans- form F ( s ). 8. Use the deﬁnition of the Laplace transform to ﬁnd the Laplace transform of f ( t ) = t +2. Problems 9-17: solve the following initial value problems using Laplace transforms.

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Unformatted text preview: 9. y 00 + 3 y + 2 y = 0, y (0) =-1, y (0) = 3 10. y 00 + y-6 y = e t , y (0) = 0, y (0) = 0 11. y 00 + y-6 y = e 2 t , y (0) = 0, y (0) = 0 1 12. y 00-4 y + 4 y = 6, y (0) = 2, y (0) = 2 13. y 00-2 y + 10 y = 3 e t , y (0) =-1, y (0) = 0 14. y 00 + 3 y + 2 y = 2 + u 2 ( t ) e-t , y (0) = 3, y (0) =-2 15. y 00 + 4 y = ± 1 if t < π cos t if t ≥ π , y (0) = 0, y (0) = 0. 16. y 00 + y-6 y = e-t + 3 δ ( t-1), y (0) = 1, y (0) = 2. 17. y 00-2 y + 2 y = 2 δ ( t-4), y (0) = 3, y (0) = 1. 18. If possible, compute the following expressions, using A = ² 1-1 2 3-1 ³ , B = 2 1-3 2-2 3 , C = ² 1 1 1 1 2 1 ³ . (a) AB (b) BA (c) AC (d) A + B (e) A + C (f) A + C T (g) B + C T 2...
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exam3sampleq - 9 y 00 3 y 2 y = 0 y(0 =-1 y(0 = 3 10 y 00...

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