6.021/Notes/2006-09-07

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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 = yhomo + ynonhomo

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 y0, final value y_\infty, time constant τ.

Example problems

RC circuit, water tank, flux across cell

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