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)}

(a) Take the Laplace transform of both sides:

where the transform of comes from

This yields the linear ODE,

Divides both sides by :

Find the integrating factor:

Multiply both sides of the ODE by :

The left side condenses into the derivative of a product:

Integrate both sides and solve for :

(b) Taking the inverse transform of both sides gives

I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that is one solution to the original ODE.

Substitute these into the ODE to see everything checks out: