The given differential equation is y'' = 39x.
To determine the value of A, we can integrate the equation twice. The first integration will give us the general solution, and then we can compare it to the given form to determine the value of A.
Integrating the equation once, we get:
y' = ∫(39x) dx
y' = (39/2)x^2 + C1
Integrating again, we obtain:
y = ∫((39/2)x^2 + C1) dx
y = (39/6)x^3 + C1x + C2
Comparing this to the given general solution y = Ax^3 + C1x + C2, we can equate the coefficients:
A = 39/6
A = 6.5
Therefore, the value of A is 6.5.
Regarding the type of differential equation, the given equation y'' = 39x is a second-order linear homogeneous ordinary differential equation. It is not separable, exact, or Bernoulli because it does not meet the criteria for those specific types of differential equations.