Describing equations

Describe the following differential equations. Are they -

an ODE or PDE?
first order or higher?
linear or non-linear?
heterogeneous or homogeneous?
separable or non-separable?

a) The oscillation of a non-linear driven pendulum,

\begin{equation} \frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = -\frac{g}{l}\sin(\theta) + C\cos(\theta)\sin(\sigma t), \end{equation}

where $l$ and $\sigma$ are constant parameters and $g$ is the acceleration due to gravity.

b) The one dimensional diffusion equation,

\begin{equation} \frac{\partial T}{\partial t} = D\frac{\partial ^2 T}{\partial x^2}. \end{equation}

c) The motion of mass $m_1$ in the gravitational field of mass $m_2$ and with a viscous friction term,

\begin{equation} m_1 \frac{\mathrm{d}^2 \mathbf{r}}{\mathrm{d} t^2} = -\frac{G m_1 m_2}{r^2} \mathbf{r} - cv^2\mathbf{v}, \end{equation}

where $v$ is the velocity.

Show answer

a) This equation is a second order, linear, heterogeneous, non-separable ODE.

b) This equation is a second order, linear, homogeneous PDE. A linear, homogeneous PDE is separable and can be solved using the Separation of Variables.

c) This equation is a second order, non-linear, heterogeneous, non-separable ODE.