Definition of a non-linear first order Partial differential equation Actually I am a little bit confused about the definition. I have read two three articles but I could not find out what type of equations are called a non-linear partial differential equation. Articles are following.
https://en.wikiversity.org/wiki/Partial_differential_equations
https://www.slideshare.net/jayanshugundaniya9/advanced-engineering-mathematics-first-order-nonlinear-partial-differential-equation-its-applications
https://mat.iitm.ac.in/home/sryedida/public_html/caimna/pde/forth/forth.html
$pq = 0$ will be a first order non linear Partial differential equation? p,q are usual notation in PDE.
Please don' downvote. I know it is a silly question. But I am really confused. Please help me. I am looking forward to ur reply.
 A: A nonlinear pde is a pde in which  either the desired function(s) and/or their derivatives have either a power $\neq 1$ or is contained in some nonlinear function like $\exp, \sin$ etc, or the coordinates are nonlinear.  for example, if $\rho:\mathbb{R}^4\rightarrow\mathbb{R}$ where three of the inputs are spatial coordinates, then an example of linear: $$\partial_t \rho = \nabla^2\rho$$ and now for nonlinear nonlinear $$\partial_t \rho = \nabla^2\rho+ \cos\rho$$
As I stated at the beginning A nonlinear pde can also be a pde in which the coordinates are non linear. Example:: $$\partial_t \rho(x,y,z,t)= \nabla^2 \rho(x,y,z,t)+xy-yz $$ the $xy$ and $yz$ make it nonlinear. P and q are analogous to x y z and/or t. 
$$
\partial_t \rho(x,y,z,t)= \nabla^2 \rho(x,y,z,t)+x^{\frac{13}{21}}$$
Is also nonlinear. 
In your notation, Example:: $$\partial_t \rho(p,q)= \nabla^2 \rho(p,q)+pq$$ is nonlinear due to $pq$
A: Here's an example for finding geodesics on an arbitrary manifold:
$$\ddot{x}^k=-\Gamma_{ij}^k\dot{x}^i\dot{x}^j.$$
A: What you are looking for are functions $F(x,y,z,p,q)$ that are non-linear in $p$ and $q$ and that define a first order partial differential equation via
$$
0=F(x,y,u(x,y),u_x(x,y),u_y(x,y)).
$$
The usual trick is to establish the Lagrange-Charpit equations 
$$
ds=\frac{dx}{F_p}=\frac{dy}{F_q}=\frac{dz}{pF_p+qF_q}=-\frac{dp}{F_x+pF_z}=-\frac{dq}{F_y+qF_z}.
$$
for the characteristic curves and assemble a family of them to form the solution surface $z=u(x,y)$.
