Solve the Initial-Value Problem $\quad x\frac{\partial u}{\partial x}+ u\frac{\partial u}{\partial y}\,= u + 2x^2$

Question

Solve the initial value problem: $$\,$$ $$ x\;\frac{\delta u}{\delta x}\,+\, u\;\frac{\delta u}{\delta y}\,=\, u + 2x^2,\qquad \mathrm{ with\;initial\, conditions}:\,u\,(x,\,\frac{1}{4}-x^2)=x$$

$$\,$$

This is what i've done so far, but I need help with this

1. Parameterize the initial conditions: $\quad x=x_0(s),\qquad y=\frac{1}{4}-s^2,\qquad u=u_0(s)$
2. Set of equations of characteristics: $\quad 1)\;\frac{dx}{d\tau}=x\qquad 2)\;\frac{dy}{d\tau}=u\qquad 3)\;\frac{du}{d\tau}=u+2x^2\qquad$

I don't know if this is even correct? $\;$Thank you