Answer to Question #200427 in Differential Equations for Ayush

Question

Equation 

x  y p  x  y q  z  z

2

cos( ) sin( )

is a quasi-linear equation

Answer 1

Quasi-linear partial differential equations of first order can be written as

$$P(x,y,z)p+Q(x,y,z)q=R(x,y,z)\quad\quad (1)$$

where $P,Q,R$ are functions of $x,y$ and $z$ . They do not involve $p$ or $q$ . Here, linear means that $p$ and $q$ appear to the first degree only.

In our case, the equation

$$\cos(x+y)p+\sin(x+y)q=z^2$$

is a quasi-linear equation.

 It's linear with respect to the derivatives (p and q)