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