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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]


  1. Find homogenous solution first
  2. 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