Let's say we have quasi-linear first order PDE
$$a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u)$$
Solving it, is equivalent to finding first integrals of following ODE system
$$\frac{\partial x}{\partial t}=a(x(t),y(t),u(t))\hspace{5pt}\frac{\partial y}{\partial t}=b(x(t),y(t),u(t))\hspace{5pt}\frac{\partial u}{\partial t}=c(x(t),y(t),u(t)).$$
Given two independent first itegrals $\Phi,\Psi$ we have general solution of our equation given by $F(\Psi,\Phi),$ where $F$ is arbitrary $C^1$ function.
In practice however we write something called Lagrange-Charpit equations
$$\frac{dx}{a(x,y,u)}=\frac{dy}{b(x,y,u)}=\frac{dz}{c(x,y,u)}$$
and use some algebraic wishy washy methods to find $\Phi,\Psi.$
I am aware of two methods to derive $\Phi,\Psi.$
First. Take one of element of Lagrange-Charpit equations, lets say $\frac{dx}{a(x,y,u)}=\frac{dy}{b(x,y,u)}.$ Now using algebraic methods, if possible transform it to $A(x)dx-B(y)dy=0.$ As a result
$$\int A(x)dx-\int B(y)dy$$
is first integral.
Second. If maneuvering with Lagrange-Charpit equations we can find a function $G$ such that $dG=0,$ then $G$ is a first integral.
Also I heard something about finding integrating factor $\mu.$
Question 1 Are there any other methods of finding first integral (using Lagrange-Charpit equations) . If so, can you write it with an example or give me some reference?
I am asking this question because I have problem with following example. Namely in book "PDE throu examples and exercises" of Pap and Takaci I encountered the example
$$xu_x+(z+u)u_y+(y+u)u_z=(y+z)$$
Authors gives following explenation
Unfortunetely I understand nothing.
Best Answer
Let $$\frac{d y}{z + u} = \frac{d z}{y + u} = \frac{d u}{y +z}.$$ Then,
$$ \frac{dy}{z + u} - \frac{dz}{y + u} = 0, \qquad \frac{dz}{y + u} - \frac{du}{y + z} = 0, \qquad \frac{du}{y + z} - \frac{dy}{z + u} = 0 $$ so \begin{array}\, f &= (y + u)dy - (z + u) dz &= 0,\\ g &= (y + z)dz - (y + u) du &= 0,\\ h &= (z + u)du - (y + z) dy &= 0. \end{array} From here, is easy to see that $$ \frac{g + h}{y^2 - z^2} - \frac{h + f}{z^2 - u^2} = 0 $$ is identity you are looking.