theory for separation of variables
The first order (http://planetmath.org/ODE) ordinary differential equation where one can separate the variables has the form where may be expressed as a product or a quotient of two functions (http://planetmath.org/ProductOfFunctions), one of which depends only on and the other on . Such an equation may be written e.g. as
We notice first that if has real zeroes (http://planetmath.org/ZeroOfAFunction) , then the equation (1) has the constant solutions and thus the lines are integral curves. Similarly, if has real zeroes , one has to include the lines to the integral curves. All those lines the -plane into the rectangular regions. One can obtain other integral curves only inside such regions where the derivative attains real values.
Let be such a region, defined by
Let be an integral curve passing through the point of the region . By the above assumptions, the derivative maintains its sign on the curve so long is inside , which is true on a neighbourhood of , contained in . This implies that as runs the interval , it defines the ordinate of uniquely as a monotonic function which satisfies the equation (1):
The last equation may be written
Because is continuous and monotonic on the interval , it can be taken as new variable of integration (http://planetmath.org/SubstitutionForIntegration) in the left hand side of (3): substitute , and change the to and .
Accordingly, the equality
is valid, meaning that if an integral curve of (1) passes through the point , the integral curve is represented by the equation (4) as long as the curve is inside the region .
Additionally, it is possible to justificate that if is an interior point of a region where and are real, continuous and , then one and only one integral curve of (1) passes through this point, the curve is regular (http://planetmath.org/RegularCurve), and both and are monotonic on it. N.B., the Lipschitz condition for the right hand side of (1) is not necessary for the justification.
When the point changes in the region , (4) gives a family of integral curves which cover the region once. The equations of these curves may be unified to the form
and integration of both sides.
- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||theory for separation of variables|
|Date of creation||2013-03-22 18:37:43|
|Last modified on||2013-03-22 18:37:43|
|Last modified by||pahio (2872)|