Answer to Question #108007 in Differential Equations for Nikesh gautam pandit ji

Question

State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example.
i) Equation cos(x+y)p + sin(x+y)q = z^2
is a quasi-linear equation.
ii) The solution of PDE
dz/dx + dz/dy =z^2 is z = -[y+f(x-y)]

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

Answer 1

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

i) Equation cos(x+y)p + sin(x+y)q = z^2

is a quasi-linear equation.

ii) The solution of PDE

dz/dx + dz/dy =z^2 is z = -[y+f(x-y)]

Iii)(dz/dx)(dz/dy) - (dz/dy)^2 = 0

is a non-linear PDE.

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

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

$z_x+z_y=-1 {=}\mathllap{/\,} z ^ 2$

False

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