# derivatives of solution of first order ODE

Suppose that $f$ is a continuously differentiable function defined on an open subset $E$ of ${\mathbb{R}}^{2}$, i.e. it has on $E$ the continuous^{} partial derivatives^{} ${f}_{x}^{\prime}(x,y)$ and ${f}_{y}^{\prime}(x,y)$.

If $y(x)$ is a solution of the ordinary differential equation^{}

$\frac{dy}{dx}}=f(x,y),$ | (1) |

then we have

${y}^{\prime}(x)=f(x,y(x)),$ | (2) |

${y}^{\prime \prime}(x)={f}_{x}^{\prime}(x,y(x))+{f}_{y}^{\prime}(x,y(x)){y}^{\prime}(x)$ | (3) |

(see the http://planetmath.org/node/2798general chain rule^{}). Thus there exists on $E$ the second derivative ${y}^{\prime \prime}(x)$ which is also continuous. More generally, we can infer the

Theorem. If $f(x,y)$ has in $E$ the continuous partial derivatives up to the order $n$, then any solution $y(x)$ of the differential equation (1) has on $E$ the continuous derivatives ${y}^{(i)}(x)$ up to the order (http://planetmath.org/OrderOfDerivative) $n+1$.

Note 1. The derivatives ${y}^{(i)}(x)$ 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:

$${y}^{\prime \prime \prime}={f}_{xx}^{\prime \prime}+2{f}_{xy}^{\prime \prime}{y}^{\prime}+{f}_{yy}^{\prime \prime}{y}^{\prime 2}+{f}_{y}^{\prime}{y}^{\prime \prime},$$ |

$${y}^{(4)}={f}_{xxx}^{\prime \prime \prime}+3{f}_{xxy}^{\prime \prime \prime}{y}^{\prime}+3{f}_{xyy}^{\prime \prime \prime}{y}^{\prime 2}+{f}_{yyy}^{\prime \prime \prime}{y}^{\prime 3}+3{f}_{xy}^{\prime \prime}{y}^{\prime \prime}+3{f}_{yy}^{\prime \prime}{y}^{\prime}{y}^{\prime \prime}+{f}_{y}^{\prime}{y}^{\prime \prime \prime}$$ |

Note 2. It follows from (3) that the curve

${f}_{x}^{\prime}(x,y)+{f}_{y}^{\prime}(x,y)f(x,y)=\mathrm{\hspace{0.33em}0}$ | (4) |

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.

## References

- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title | derivatives of solution of first order ODE |
---|---|

Canonical name | DerivativesOfSolutionOfFirstOrderODE |

Date of creation | 2013-03-22 18:59:14 |

Last modified on | 2013-03-22 18:59:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 34A12 |

Classification | msc 34-00 |

Related topic | SolutionsOfOrdinaryDifferentialEquation |

Related topic | InflexionPoint |