6.021/Notes/2006-09-07

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Differential Equations

First order: no terms of order higher than [math]\frac{dy}{dt}[/math] (such as [math]\frac{d^2y}{dt^2}[/math])

Linear: no product of dependent variables such as [math]y\cdot\frac{dy}{dt}[/math]

Homogenous: [math]y=0[/math] is a solution

Solving first-order linear equations

General form: [math]\alpha\frac{dy}{dt}+\beta y = \gamma \rightarrow \tau\frac{dy}{dt}+y=y_\infty[/math]

Solving:

  1. Find homogenous solution first
  2. Assume solution is [math]y=y_{homo}+y_{non-homo}[/math]

General solution: [math]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]y_0[/math], final value [math]y_\infty[/math], time constant [math]\tau[/math].

Example problems

RC circuit, water tank, flux across cell