Consider the differential equation: 2y'' + ty' − 2y = 14, y(0) = y'(0) = 0.

In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients.

If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . ,then
ℒ{tnf(t)} = (-1)^n d^n/ds^n F(s)

to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = ℒ{y(t)}.

Requried:
a. Sovle the first order DE for Y(s).
b. Find find y(t)= ℒ^-1 {Y(s)}