method of integrating factors
If there is a solution of (1) which may be expressed in the form
It’s straightforward to show that if is a non-constant solution of the equation (2), then all solutions of this equation are where is a freely chosen function with (mostly) continuous derivative.
The connection of the equations (1) and (2) may be presented also in another form. Suppose that is any solution of (1). Then (2) implies the proportion equation
If we denote the common value of these two ratios by , then we have
This gives to the differential of the function the expression
We see that is the integrating factor or Euler multiplicator of the given differential equation (1), i.e. the left hand side of (1) turns, when multiplied by , to an exact differential (http://planetmath.org/ExactDifferentialForm).
Conversely, any integrating factor of (1), i.e. such that is the differential of some function , is easily seen to determine the solutions of the form of (1). Altogether, solving the differential equation (1) is equivalent with finding an integrating factor of the equation.
When an integrating factor of (1) is available, the solution function can be gotten from the line integral
along any curve connecting an arbitrarily chosen point and the point in the region .
Note. In general, it’s very hard to find a suitable integrating factor. One special case where such can be found, is that and are homogeneous functions of same degree (http://planetmath.org/HomogeneousFunction): then the expression is an integrating factor.
Example. In the differential equation
we see that and both define a homogeneous function of degree (http://planetmath.org/HomogeneousFunction) 4. Thus we have the integrating factor , and the left hand side of the equation
is an exact differential. We can integrate it along the broken line, first from to and then still to , obtaining
So the general solution of the given differential equation is
- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||method of integrating factors|
|Date of creation||2013-03-22 16:31:48|
|Last modified on||2013-03-22 16:31:48|
|Last modified by||pahio (2872)|