The order and degree of the following differential equation are $m$ and $n$, respectively.

$$ \frac{\partial^3 \varphi}{\partial x^3}+\frac{\partial^2 \varphi}{\partial^2 y^2} \frac{\partial \varphi}{\partial x}+\left(\frac{\partial^2 \varphi}{\partial x^2}\right)^2+\frac{\partial \varphi}{\partial y}=0 $$

The value of $(m-n)$ is

For the differential equation given below, which one of the following options is correct?

$$ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,0 \leq x \leq 1,0 \leq y \leq 1 $$

Solution of ∇²T = 0 in a square domain (0 < x < 1 and 0 < y < 1) with boundary conditions:

T(x, 0) = x; T(0, y) = y; T(x, 1) = 1 + x; T(1, y) = 1 + y is

The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1$$ and $${\left. {{{dy} \over {dx}}} \right|_{x = {\pi \over 2}}} = - 1$$ has