A differential equation is an equation for a function containing derivatives of that function. For example,
the differential equations for an RLC circuit, a pendulum, and a diffusing dye.
The RLC circuit equation (and pendulum equation) is an ordinary differential equation, or ode, and the diffusion equation is a partial differential equation, or pde. An ode is an equation for a function of a single variable and a pde for a function of more than one variable. A pde is theoretically equivalent to an infinite number of odes, and numerical solution of nonlinear pdes may require supercomputer resources.
The RLC circuit and the diffusion equation are linear and the pendulum equation is nonlinear. In a linear differential equation, the unknown function and its derivatives appear as a linear polynomial.
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