19
edits
Changes
Jump to navigation
Jump to search
Line 22:
Line 22: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Differential equation (view source)
Revision as of 06:01, February 27, 2009
, 06:01, February 27, 2009no edit summary
Obviously, partial differential equations are much more complicated to solve.
Obviously, partial differential equations are much more complicated to solve.
==Methods==
There are many ways to find solutions to differential equations.
===Ordinary Differential Equations===
The simplest differential equations to solve are ''separable'' differential equations. A differential equation is separable if it can be written in the form
<math> \frac{d^{(n)}y}{dx^{(n)}} = F(x)G(y) </math>
Then we can separate the two variables, collect the x's on one side and the y's on the other side, then integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found the function y. For example, for the derivative n = 1:
<math> \frac{dy}{dx} = F(x)G(y)
\frac{dy}{G(y)} = F(x)dx
\int\frac{dy}{G(y)} = \int F(x)dx </math>
The solution is then given implicitly by the expression:
<math>\int\frac{dy}{G(y)} - \int F(x)dx = C </math>
where C is an arbitrary constant.
====Linear Differential Equation Solutions====
----
If a differential equation can be written in the form
<math>F_n(x)\frac{d^{(n)}y}{dx^{(n)}} + F_{n-1}(x)\frac{d^{(n-1)}y}{dx^{(n-1)}} + F_{n-2}(x)\frac{d^{(n-2)}y}{dx^{(n-2)}} + ... + F_1(x)\frac{dy}{dx} + F_0(x)y = G(x)</math>
it is considered a ''linear'' differential equation.
=====First Order Linear Equations=====
The ''order'' of a differential equation is equal to the degree of the highest derivative in the equation. For example, the above equations are order n equations. A first order linear equation appears in the form:
<math> \frac{dy}{dx} + F(x)y = G(x) </math>
==References==
==References==