value problems for second-order linear differential equations with constant coefficients. Many Sections in this chapter are numerous initial- can be interpreted as models of particular physical systems, but usually we do not point this out explicitly. Problems slogany In each of Problems 1 through 7, find the inverse Laplace transform of the given function. 3 1. F(s) = 2. F(s) = 3. F(s) = 4. F(s) = 5. F(s) = 6. F(s) = 7. F(s) $² = +4 4 85²-4s + 12 s(s² + 4) 1-2s s² + 4s +5 In each of Problems 8 through 16, use the Laplace transform to solve the given initial value problem. (s - 1)³ 2 s² + 35-4 2s+2 s²+2s +5 2s-3 52-4 imonylag y(0) = 1, y'(0) = -1 8. y" - y'-6y=0; 9. y" + 3y' +2y=0; y(0) = 1, y'(0) = 0 (@))2 10. y" - 2y + 2y = 0; 11. y" - 2y + 4y = 0; y(0) = 0, y'(0) = 1 y(0) = 2, 12. y" +2y + 5y = 0; y(0) = 2, 13. y(4) - 4y"" + 6y" - 4y' + y = 0; y'(0) = 1, y"(0) = 0, y""(0) = 1 14. y(4) - y = 0; y(0) = 1, y'(0) = 0, y" (0) = 1, y"" (0) = 0 15. y"+w²y = cos(2t), w² #4; y(0) = 1, y'(0) = 0 da 16. y" - 2y' +2y=et; y(0) = 0, y'(0) = 1 In each of Problems 17 through 19, find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem. A method of determining the inverse transform is developed in Section 6.3. You may wish to refer to Problems 16 through 18 in Section 6.1. 17. y" +4y= [1, 0≤t<,vis ad to motanim non y(0) = 1, y'(0) = 0 π ≤t< 00; 0, ft, 0≤t < 1, 1, 1≤t<∞0; 18. y" + 4y = t, amaldor y'(0) = 0 motanen sosiqa y'(0) = -1 y(0) = 0, 19. y"+y=2-t, 0, y(0) = 0, y'(0) = 0 0 ≤ t < 1, 1st <2, y(0) = 0, y'(0) = 0 2 ≤t <∞0; ad