PDEs satisfied by implicitly defined functions

Prove that the function $z=z(x,y)$ defined by the equation

[where $\Phi (u,v)$ is an arbitrary differentiable function of the variables $u$, $v$, and $a$ and $b$ are constants] is a solution of the equation

Describe the geometric nature of the surface (1).

Prove that the function $z=z(x,y)$ defined by the equation

[where $\Phi (u,v)$ is an arbitrary differentiable function of the variables $u$ and $v$] satisfies the equation

Describe the geometric nature of the surface (2).

Prove that the function $z=z(x,y)$ defined by the equation

[where $\Phi (u)$ is an arbitrary differentiable function of the variable $u$, and $a$, $b$ and $c$ are constants] satisfies the equation

Describe the geometric nature of the surface (3).

The function $z=z(x,y)$ is given by the equation

Prove that:

The function $z=z(x,y)$ is given by the equation

Prove that:

Prove that the function $z=z(x,y)$ defined by the system of equations

[where $\alpha =\alpha (x,y)$ is a parameter and $f(\alpha )$ is an arbitrary differentiable function] satisfies the equation

Prove that the function $z=z(x,y)$ given by the system of equations

satisfies the equation

Prove that the function $z=z(x,y)$ defined by the system of equations

satisfies the equation

Prove that the function $z=z(x,y)$ given by the system of equations

satisfies the equation

Prove that the implicit function $z=z(x,y)$ defined by the equation $y=x\varphi (z)+\psi (z)$ satisfies the equation

Prove that if the equation $f\left(x^{2}+y^{2}+z^{2}\right) = a x+y b+c z$, where $f(u)$ is a differentiable function and $a, b, c$ are constants, defines a differentiable function $z(x ; y)$, then it satisfies the equation

Prove that if the equation $y f(z / y) = x^{2}+y^{2}+z^{2}$, where $f(u)$ is a differentiable function, defines a differentiable function $z(x ; y)$, then it satisfies the equation

Prove that if the equation $f(x-a z ; y-b z) = 0$, where $f(u ; v)$ is a differentiable function and $a, b$ are constants, defines a differentiable function $z(x ; y)$, then it satisfies the equation

Prove that if the equation $f\left(\frac{x-a}{z-c} ; \frac{y-b}{z-c}\right) = 0$, where $f(u ; v)$ is a differentiable function and $a, b, c$ are constants, defines a differentiable function $z(x ; y)$, then it satisfies the equation

Prove that if the equation $f\left(\frac{x_{1}}{x_{n}} ; \frac{x_{2}}{x_{n}} ; \ldots ; \frac{x_{n-1}}{x_{n}} ; \frac{z}{x_{n}^{\alpha }}\right) = 0$, where $f\left(u_{1} ; u_{2} ; \ldots ; u_{n}\right)$ is a differentiable function and $\alpha$ is a constant, defines a differentiable function $z\left(x_{1} ; x_{2} ; \ldots ; x_{n}\right)$, then it satisfies the equation

Let the system of equations

where $f(v)$ is a twice differentiable function, define differentiable functions $u(x ; y)$ and $v(x ; y)$. Prove that