Can anyone good at math solve ANOTHER question?

Question

Concept: Solving Differential Equations

Let (d^2y)/(dx^2) =6x.

a) find a solution to the differential equation that is continuous for -infinity < x
b) How many functions like this are there? How do you know?

Thank you SO MUCH in advance! :)

Answer 1

This initial value problem is surprisingly simple, since niether y nor any derivatives other than the second appear in the equation, so all that is necessary is to integrate twice:
∫∫d²y/dx² dx dx = ∫∫6x dx dx
∫dy/dx dx = ∫3x² + C dx
y=x³+Cx+K

Now we need merely find the constants C and K. We know that y'(0)=3(0)² + C = 0, so C=0. Further, we know that y(0) = K = 1, so K=1.

Thus y=x³+1

b: There is exactly one such function. All antiderivatives of a function are the same up to addition of an arbitrary constant, and as we showed when solving the problem, there is only one consistent set of values for the two constants of integration introduced in this problem.