Generally, the general form of the partial differential equation in quasilinear form is given as
$$Au_{xx}+Bu_{xy}+Cu_{yy} = G(u, u_x, x,y,u_y)$$
In the above, $G$ is a function of mentioned variables and here $u_{xx}$ denotes $\frac{\partial^2 u}{\partial x^2}$ and $u = u(x,y)$. I have found the characteristic curve slope for above equation using transformation variables $\psi$ and $\eta$ each being function of $(x,y)$ and have written the above equations as follows
$$A^*u_{xx}+B^*u_{xy}+C^*u_{yy} = G^*(u, u_x, x,y,u_y)$$
Here,
\begin{align} A^* &= A(\psi_x)^2 + B(\psi_x)(\psi_y)+ C(\psi_y)^2 \\ C^* &= A(\eta_x)^2 + B(\eta_x)(\eta_y)+ C(\eta_y)^2 \\ B^* &= 2(A\psi_x\eta_x + C\psi_y\eta_y)+ B(\psi_x\eta_y + \psi_y\eta_x) \end{align}
Now, my transformation coordinates must be such that $A^*$ and $C^*$ go to zero. Also since they both have same form, I can introduce a new variable $\theta$ such that it accounts for $\psi$ and $\eta$. Therefore we can write $A^*$ and $C^*$ combined as
$$A(\theta_x)^2 + B(\theta_x)(\theta_y)+ C(\theta_y)^2 = 0 \implies A \left(\frac{\theta_x}{\theta_y} \right)^2 + B\frac{\theta_x}{\theta_y} + C = 0 \tag{1}$$
Now, $\theta(x,y) = \text{constant}$, therefore, $d\theta=0$. Hence we can write
$$\frac{dy}{dx} = -\frac{\theta_x}{\theta_y}$$ which is basically the slope of the characteristic curve. From $(1)$, using slope of characteristic curve, we can write
$$\frac{dy}{dx} = \frac{B \pm \sqrt{B^2 - 4AC}}{2A} \tag{2}$$
where we define the discriminant as $\Delta = \sqrt{B^2 - 4AC}$.
My doubt is basically, how to find for the hyperbolic case when $\Delta > 0$. After integrating $(2)$ we get the following
$$y = \frac{B \pm \sqrt{B^2 - 4AC}}{2A}x + c$$
but now, how to convert above $y$ to $\psi$ and $\eta$ again? How can I treat the integration constant and why should I do so, as transformed variables, $i.e$ $\psi, \eta$?
Attached is the referring document with this question. Thank you for your help.
