Question

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.

(1) Equation cos(x+y) p+sin(x+y) q=z^2+z is a quasi -linear equation.

(2) The solution of PDE dz/dx+dz/dy=z^2 is z=-[y+f(x-y) ].

(3) dz/dx.dz/dy-(dz/dy) ^2=0is a non -linear PDE.

Answer 1

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

2)$z_x=-f'(x-y);z_y=-1+f'(x-y)$

$z_x+z_y=-1\ne z^2$

False

3)True. It contains a nonlinear element $(\frac{dz}{dy})^2$