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

General form: $\alpha\frac{dy}{dt}+\beta y = \gamma \rightarrow \tau\frac{dy}{dt}+y=y_\infty$

Solving:

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 $τ$ .

RC circuit, water tank, flux across cell