# second order ordinary differential equation

A second order ordinary differential equation$F(x,\,y,\,\frac{dy}{dx},\,\frac{d^{2}y}{dx^{2}})=0$  can often be written in the form

 $\displaystyle\frac{d^{2}y}{dx^{2}}\;=\;f\left(x,\,y,\,\frac{dy}{dx}\right).$ (1)

By its general solution one means a function$x\mapsto y=y(x)$  which is at least on an interval twice differentiable and satisfies

 $y^{\prime\prime}(x)\;\equiv\;f(x,\,y(x),\,y^{\prime}(x)).$

By setting  $\frac{dy}{dx}:=z$,  one has  $\frac{d^{2}y}{dx^{2}}=\frac{dz}{dx}$,  and the equation (1) reads  $\frac{dz}{dx}=f(x,\,y,\,z)$.  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

 $\displaystyle\begin{cases}\frac{dy}{dx}=z,\\ \frac{dz}{dx}=f(x,\,y,\,z),\end{cases}$ (2)

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

 $\displaystyle\begin{cases}\frac{dy}{dx}=\varphi(x,\,y,\,z),\\ \frac{dz}{dx}=\psi(x,\,y,\,z),\end{cases}$ (3)

where $y$ and $z$ are unknown functions of the variable $x$.  The existence theorem of the solution

 $\displaystyle\begin{cases}y=y(x),\\ z=z(x)\end{cases}$ (4)

is as follows; cf. the Picard–Lindelöf theorem (http://planetmath.org/PicardsTheorem2).

Theorem.  If the functions $\varphi$ and $\psi$ are continuous and have continuous partial derivatives with respect to $y$ and $z$ in a neighbourhood of a point  $(x_{0},\,y_{0},\,z_{0})$,  then the normal system (3) has one and (as long as $|x\!-\!x_{0}|$ does not exceed a certain ) only one solution (4) which satisfies the initial conditions$y(x_{0})=y_{0},\;\,z(x_{0})=z_{0}$.  The functions (4) are continuously differentiable in a neighbourhood of $x_{0}$.

## References

• 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title second order ordinary differential equation SecondOrderOrdinaryDifferentialEquation 2013-03-22 18:35:39 2013-03-22 18:35:39 pahio (2872) pahio (2872) 5 pahio (2872) Topic msc 34A05 normal system normal system of second order