numerical solutions/uniqueness of a D.E

ok

I have two equations;

(1) d^2y/dx^2 + x^2.y^2 = x, y(0) = y(2) = 0

(2) d^2y/dx^2 + x^2.y^2 = x, y(0) = 0, y'(0) = -0.7

I cant solve these algebraically since they are non-linear, solving numerically and plotting a solution, I find that these graphs are very similar for x between 0 and 2. (a parabola going through (0,0) and (2,0), with a minimum at about -0.4)

does this mean that the solution to (1) has the property y'(0)= -0.7?

so are the differential equations equivalent since a solution can be described uniquely by ivps or boundary conditions/

many thanks

Re: numerical solutions/uniqueness of a D.E

Nope. The solutions to these two problems are not the same.

The first problem is a boundary value problem, meaning the solution is defined at the two endpoints x=0 and x=2 of the domain of definition.

The second problem is an initial value problem, where the values for y and y' are fixed at the initial point x=0.

The first problem is likely to have a periodical solution, the second most surely not.

Anyway, its a lazy Sunday, I trust theis info is enough.