Classification of differential equations

Classify the following equations. Are they ODEs or PDEs? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant-coefficient? If it is an ODE\@, is it autonomous?

1. $\displaystyle \frac{\partial^2 v}{\partial x^2} + 3 \frac{\partial^2 v}{\partial y^2} = \sin(x)$
2. $\displaystyle \frac{d x}{dt} + \cos(t) x = t^2+t+1$
3. $\displaystyle \frac{d^7 F}{dx^7} = 3F(x)$
4. $\displaystyle y''+8y'=1$
5. $\displaystyle x''+tyx'=0, \quad y''+txy = 0$
6. $\displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial s^2} + u^2$

Write down the general zeroth-order linear ordinary differential equation. Write down the general solution.

For which $k$ is $\frac{dx}{dt}+x^k = t^{k+2}$ linear? Hint: There are two answers.