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#+title: differential equation
#+author: Preston Pan
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* Introduction
A differential equation is an equation whose solutions are functions and which incorporate derivatives of the function
you're solving for. Differential equations often have a family of infinite solutions, where a general solution for
a differential equation incorporates many particular solutions. Particular solutions to differential equations are specific
functions, corresponding to a single choice of an initial value problem. Therefore, general solutions tell you how to solve
initial value problems.
Differential equations are often used to model real world systems, and are the main tool in numerical simulations of said
systems.
** ODE
An ODE is a differential equation involving single variable function solution classes and their derivatives.
** PDE
A PDE is a differential equation involving multivariable function solution classes and generally involve partial derivatives
of the unknown function.
** initial value problem
An initial value problem is a problem where one is given a differential equation and particular values of the unknown function
and particular values of its derivatives, and the result is a particular solution.
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