6.021/Notes/2006-09-07

Differential Equations
First order: no terms of order higher than $$\frac{dy}{dt}$$ (such as $$\frac{d^2y}{dt^2}$$)

Linear: no product of dependent variables such as $$y\cdot\frac{dy}{dt}$$

Homogenous: $$y=0$$ is a solution

Solving first-order linear equations
General form: $$\alpha\frac{dy}{dt}+\beta y = \gamma \rightarrow \tau\frac{dy}{dt}+y=y_\infty$$

Solving:
 * 1) Find homogenous solution first
 * 2) Assume solution is $$y=y_{homo}+y_{non-homo}$$

General solution: $$y(t)=(y_0-y_\infty)e^{-\frac{t}{\tau}}+y_\infty$$

Only three things needed for all such 1st order linear equations: initial value $$y_0$$, final value $$y_\infty$$, time constant $$\tau$$.

Example problems
RC circuit, water tank, flux across cell