second order ordinary differential equation
A second order ordinary differential equation can often be written in the form
By setting , one has , and the equation (1) reads . It is easy to see that solving (1) is equivalent (http://planetmath.org/Equivalent3) with solving the system of simultaneous first order (http://planetmath.org/ODE) differential equations
the so-called normal system of (1).
The system (2) is a special case of the general normal system of second order, which has the form
where and are unknown functions of the variable . The existence theorem of the solution
is as follows; cf. the Picard–Lindelöf theorem (http://planetmath.org/PicardsTheorem2).
Theorem. If the functions and are continuous and have continuous partial derivatives with respect to and in a neighbourhood of a point , then the normal system (3) has one and (as long as does not exceed a certain ) only one solution (4) which satisfies the initial conditions . The functions (4) are continuously differentiable in a neighbourhood of .
- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||second order ordinary differential equation|
|Date of creation||2013-03-22 18:35:39|
|Last modified on||2013-03-22 18:35:39|
|Last modified by||pahio (2872)|
|Defines||normal system of second order|