ODE - Laplace Transform, Delta function, Ordinary differential equation with initial values

Équations Différentielles (2e édition revue et augmentée) - Mario Lefebvre Ch 5: Transformées de Laplace 5.4: La fonction delta de Dirac Ex 5-13: Let f(x) = (1/(sqrt{2pi}sigma)) exp(-x^2/(2sigma^2)) for real x, where sigma is a positive constant. One can show that \int_{-infinity}^{infinity} f(x) dx = 1. Use this result to show that lim_{sigma to 0} f(x) = delta(x). Ex 5-14: Calculate (a) \int_{-infinity}^{infinity} (delta(t-2) + delta(t+1))e^t dt; (b) \int_{-infinity}^{infinity} delta((t-2)(t+1))e^t dt. Ex 5-15: Solve the ordinary differential equations: (a) y''(t) - 2 y'(t) = delta(t-2), for t not negative, with y(0) = 0 and y'(0) = 1; (b) y''(t) - 4y(t) = delta(2t-1) for t not negative, with y(0) = 1 and y'(0) = 0. Ex 5-20: Find the solution y(t) which satisfy to the ordinary differential equation y''(t) + y'(t) = delta(t-1) for t not less than 0, with y(0) = 0 and y'(0) = 1.