Solve the following: $(20$pts$)$ Consider a system of PDEs described as:

$u\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}+\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}=0$

$u\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}+\frac{c^2}{}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}=0$

$u\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}+\frac{c^2}{}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}=0.$

Determine the mathematical properties of the PDEs, for different Mach number values. Mach number $M=$

$\frac{\sqrt[\phantom{2}]{u^2+v^2}}{c}.$