Trouble finding the function in terms of parameters.
Solution 1:
Instead of $\frac{d^2(y_0 - y)}{dy^2}$ does not make sense as it stands and instead use$m\ddot{x}$.
I believe you should use $\frac{d^2}{dt^2}$ instead.
You can actually do a first integral $$ m\ddot{x} = m\dot{x}\frac{d}{dx}\dot{x} = -kx + \alpha x^2 $$ or $$ m\frac{d}{dx}\frac{\dot{x}^2}{2}= -kx + \alpha x^2 $$ integrating w.r.t $x$ $$ \frac{m\dot{x}^2}{2} = -\frac{k}{2}x^2 + \frac{\alpha}{3}x^3 + C $$ Then you can use $x = y_0 - y$ etc.