# Laplace transform

Order polyno in Y(s) will be a first-order equation. Since we can solve a first-order equation, the major difficulty will be in PROBLEMS: Section 5.4 For Problems 1–14, use the Laplace transform to solve the given initial-value problem. 1. y’= 1 y(O)= 1 2. y’- y = 0 y(0) = 1 3. y’- y = e (0) = 1 4. y’+y=et w(0) = 1 5. y” = el (0) = 1 y(0) = 1 6. y” – 3y’ + 2y = 0 (0)=1 y'(0) = 1 7. y” + 2y’=4 y(O)= 1 y(0) = -4 8. y” +9y = 20e-1 (0) = 1 y(0) = 1 9. y” +9y = cos 3t y(0)=1 y (0) = -1 10. y” + 4y = 4 y(0) = 1 y'(0) = 1 11. y” – 2y’ + 5y = 0 (0) = 2 y'(O)=4 12. y” + 10y’ + 25y = 0 (0) = 0 y”0) = 10 13. y” + 3y’ + 2y = 6 y(0)=1 y’O) = 2 14. y” + y = sint (O) = 2 y'(O)=-1 For Problems 15-17, solve the given higher-order initial-value problem using the method of Laplace transforms. 15. y – y – y + y = 60′ 15. y(O) = 0 y'(0) = 0 y”(0) = 0 16. y + 2y” + y + 2y = 2 16. yo) – 3 y'(0) = -2 y”(0) = 3 17. y – y = 0 17. yo) = 1 y'(0) = 0 y”(0) = -1 y(0) = 0 Finding Laplace Transforms by Taylor Series Often a Laplace transform can be found by taking the Laplace transform of each term in the Taylor series of the function PROBLEMS: Section 5.5 For Problems 1–4, write each of the given functions in terms of the unit step function and sketch their graphs. a (Ost

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