View source for Solutions of Differential Equations

A solution of a differential equation is an expression of the dependent variable that satisfies the relation established in the differential equation. For example, the solution of <math> y' + y - 2 = 4x </math> will be some equation y = f(x) such that y and its first derivative, y', satisfy the relation <math> y' + y - 2 = 4x </math>. The general solution of a differential equation will have one or more arbitrary constants, depending on the order of the original differential equation (the solution of a first order diff. eq. will have one arbitrary constant, a second order one will have two, etc.).

Examples:
* <math> y' = 2x </math>. Through simple integration, we can calculate the general solution of this equation to be <math>y = x^2 + C</math>, where C is an arbitrary constant.
* <math> y' - y = 0 </math>. The G.S. is <math> y = Ce^{x} </math>. <math> y' = Ce^{x} </math>, so <math> y' - y = 0 \to Ce^{x} - Ce^{x} = 0</math>, so this solution satisfies the relationship for all arbitrary constants C.
* <math> y' + y - 2 = 0 </math>. The G.S. is <math> y = Ce^{-x} + 2 </math>. <math> y' = -Ce^{-x} </math>, so <math> 0 = y' + y - 2 </math> becomes <math> 0 = -Ce^{-x} + Ce^{-x} + 2 - 2 = 0 </math>.


==Resources==
* [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations],