derivatives of solution of first order ODE
If is a solution of the ordinary differential equation
then we have
Theorem. If has in the continuous partial derivatives up to the order , then any solution of the differential equation (1) has on the continuous derivatives up to the order (http://planetmath.org/OrderOfDerivative) .
Note 1. The derivatives are got from the equation (1) via succesive differentiations. Two first ones are (2) and (3), and the next two ones, with a simpler notation:
Note 2. It follows from (3) that the curve
is the locus of the inflexion points of the integral curves of (1), or more exactly, the locus of the points where the integral curves have with their tangents a contact of order (http://planetmath.org/OrderOfContact) more than one. The curve (4) is also the locus of the points of tangency of the integral curves and their isoclines.
- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||derivatives of solution of first order ODE|
|Date of creation||2013-03-22 18:59:14|
|Last modified on||2013-03-22 18:59:14|
|Last modified by||pahio (2872)|