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Homesecond order ordinary differential equation

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# 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 with solving the system of simultaneous first order 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.

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 bound) 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).

## Mathematics Subject Classification

34A05*no label found*

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