6.021/Notes/2006-09-07
Differential Equations
First order: no terms of order higher than [math]\displaystyle{ \frac{dy}{dt} }[/math] (such as [math]\displaystyle{ \frac{d^2y}{dt^2} }[/math])
Linear: no product of dependent variables such as [math]\displaystyle{ y\cdot\frac{dy}{dt} }[/math]
Homogenous: [math]\displaystyle{ y=0 }[/math] is a solution
Solving first-order linear equations
General form: [math]\displaystyle{ \alpha\frac{dy}{dt}+\beta y = \gamma \rightarrow \tau\frac{dy}{dt}+y=y_\infty }[/math]
Solving:
- Find homogenous solution first
- Assume solution is [math]\displaystyle{ y=y_{homo}+y_{non-homo} }[/math]
General solution: [math]\displaystyle{ y(t)=(y_0-y_\infty)e^{-\frac{t}{\tau}}+y_\infty }[/math]
Only three things needed for all such 1st order linear equations: initial value [math]\displaystyle{ y_0 }[/math], final value [math]\displaystyle{ y_\infty }[/math], time constant [math]\displaystyle{ \tau }[/math].
Example problems
RC circuit, water tank, flux across cell
