# 6.021/Notes/2006-09-07

## Differential Equations

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

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

Homogenous: $\displaystyle{ y=0 }$ is a solution

### Solving first-order linear equations

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

Solving:

1. Find homogenous solution first
2. Assume solution is $\displaystyle{ y=y_{homo}+y_{non-homo} }$

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

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

### Example problems

RC circuit, water tank, flux across cell