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Suppose we have a first order quasi-linear PDE $$zz_x+z_y=1$$ and that we want to derive the general solution.

First we represent the characteristic curves parametrically as $$x=x(r;s),~y(r;s),~z=z(r;s)$$ where $s$ labels the initial curve. The parametric solution is defined by the ODEs $$\frac{\partial x}{\partial r}=z,~\frac{\partial y}{\partial r}=1,~\frac{\partial z}{\partial r}=1.$$ We also consider $\frac{\partial z}{\partial x}=\frac{1}{z},~\frac{\partial y}{\partial x}=\frac{1}{z}.$

The next step is to derive the following equation $z^2=2x+c_1(s).$ But I struggle to see how one derives the equation.

I would appreciate any help or suggestion. Thank you.

johnny09
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1 Answers1

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Multiply the first equation by $2\partial z/\partial r$, then use the third equation: $$ 2z\frac{\partial z}{\partial r} = 2\frac{\partial x}{\partial r} \frac{\partial z}{\partial r} = 2\frac{\partial x}{\partial r}. $$ The left-hand side is $ \partial (z^2)/\partial r $. Now integrate both sides of this with respect to $r$ as usual: $$ z^2 = 2x+c_1(s). $$

Chappers
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