Hello ı cant write like maths notation hear sorry for that.
Find the solution of the initial value problem for the quasilinear equation
$p - z q + z = 0$ for the initial data curve
$\Gamma: x_{0} = 0, y_{0} = s, z_{0} = -2s, −\infty \leq s \leq \infty $
I found $c_{1} = (e^x).z $ But ı dont know is it true or not and ı couldn't continue from this part. Can u help me ?
$$z_x-zz_y=-z$$ You correctly found a first characteristic equation : $$ze^x=c_1$$ A second characteristic equation is : $$z-y=c_2$$ The general solution is : $$ze^x=F(z-y)\quad\implies\quad z=e^{-x}F(z-y)$$ Condition : $$z(0,y)=-2y=e^0F(-2y-y)=F(-3y)$$ The function $F$ is determined : $$F(X)=\frac23 X$$ We put in into the general solution where $X=z-y$ $$ze^x=\frac23 (z-y)$$ Solving for $z$ : $$z(x,y)=\frac{-2y}{3e^x-2}$$