Please show all works with details Solve the initial value p

Please show all works with details.

Solve the initial value problem Note that delta(t - 3) consists of a time delay and an impulse function (the impulse happens at t = 3)

Solution

Taking laplace transform on both sides

L(y`` -2y` +5y) = L(2d(t-3))

(s^2Y^2-s*y(0)-y`(0)) -2(s*Y-y(0)) +5Y = 2*exp(-3s)

s^2Y^2-1 -2(sY) +5Y = 2exp(-3s)

Y^2 + Y(5/s^2-1/s) - 2exp(-3s)/s^2

This is quadraic equation in Y.