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| − | + | == Exact Equations == | |
| + | ---- | ||
| + | |||
| + | Before we begin, a quick note on notation. Within this section, subscripts of x and y mean "partial derivative with respect to x" and "partial derivative with respect to y" respectively. So | ||
| + | |||
| + | <math> | ||
| + | M_y(x,y) | ||
| + | </math> | ||
| + | |||
| + | means the partial derivative of M(x,y) with respect to y. | ||
| + | |||
| + | Suppose, firstly, that your differential equation can be written this way: | ||
| + | |||
| + | <math> | ||
| + | M(x,y)+N(x,y)y' = 0 | ||
| + | </math> | ||
| + | |||
| + | and secondly, that | ||
| + | |||
| + | <math> | ||
| + | \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} | ||
| + | </math> | ||
| + | |||
| + | In this case*, your differential equation is said to be ''exact'', and the solution to the differential equation is '''ψ(x,y)''' where | ||
| + | |||
| + | <math> | ||
| + | \psi_x(x,y)=M(x,y);\qquad \psi_y(x,y)=N(x,y) | ||
| + | </math> | ||
Revision as of 13:21, 10 October 2009
Exact Equations
Before we begin, a quick note on notation. Within this section, subscripts of x and y mean "partial derivative with respect to x" and "partial derivative with respect to y" respectively. So
$ M_y(x,y) $
means the partial derivative of M(x,y) with respect to y.
Suppose, firstly, that your differential equation can be written this way:
$ M(x,y)+N(x,y)y' = 0 $
and secondly, that
$ \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} $
In this case*, your differential equation is said to be exact, and the solution to the differential equation is ψ(x,y) where
$ \psi_x(x,y)=M(x,y);\qquad \psi_y(x,y)=N(x,y) $
