# second order ordinary differential equation

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

$\frac{{d}^{2}y}{d{x}^{2}}}=f(x,y,{\displaystyle \frac{dy}{dx}}).$ | (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 $\frac{{d}^{2}y}{d{x}^{2}}=\frac{dz}{dx}$, 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^{}

$\{\begin{array}{cc}\frac{dy}{dx}=z,\hfill & \\ \frac{dz}{dx}=f(x,y,z),\hfill & \end{array}$ | (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

$\{\begin{array}{cc}\frac{dy}{dx}=\phi (x,y,z),\hfill & \\ \frac{dz}{dx}=\psi (x,y,z),\hfill & \end{array}$ | (3) |

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

$\{\begin{array}{cc}y=y(x),\hfill & \\ z=z(x)\hfill & \end{array}$ | (4) |

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

Theorem. If the functions $\phi $ 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 |
---|---|

Canonical name | SecondOrderOrdinaryDifferentialEquation |

Date of creation | 2013-03-22 18:35:39 |

Last modified on | 2013-03-22 18:35:39 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 34A05 |

Defines | normal system |

Defines | normal system of second order |