transform the equation in the respective region to canonical form: $u_{xx} + u_{xy} - xu_{yy} = 0$

Question

Here what i did: $$a=1 ,\quad b=1,\quad c=-x$$ $$\Delta = 1 + 4x$$ $$x < -\frac{1}{4} \rightarrow eliptic$$ $$x = -\frac{1}{4} \rightarrow parabolic$$ $$x > -\frac{1}{4} \rightarrow hyperbolic$$ Characteristic equations: $$\frac{dy}{dx} = \frac{b \pm \sqrt{b^2 - 4ac} }{2a}= \frac{1 \pm \sqrt{1 + 4x} }{2}$$ $$\rightarrow y = \frac{x}{2} + \frac{1}{2}( \frac{(4x+1)^\frac{5}{2})}{40}-\frac{(4x+1)^\frac{3}{2})}{24}) + C_1$$ $$y = \frac{x}{2} - \frac{1}{2}( \frac{(4x+1)^\frac{5}{2})}{40}-\frac{(4x+1)^\frac{3}{2})}{24}) + C_2$$ I’m stuck here and don’t know what to do next. I think with this result, the next steps will be very difficult and complicated. So, I want to ask if there is a better way to do this and if I made a mistake at any step? I look forward to receiving answers and hints from everyone. Thank you all very much