# 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

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

then we have

 $\displaystyle y^{\prime}(x)\;=\;f(x,\,y(x)),$ (2)
 $\displaystyle y^{\prime\prime}(x)\;=\;f_{x}^{\prime}(x,\,y(x))+f_{y}^{\prime}(% x,\,y(x))\,y^{\prime}(x)$ (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:

 $y^{\prime\prime\prime}\;=\;f_{xx}^{\prime\prime}+2f_{xy}^{\prime\prime}y^{% \prime}+f_{yy}^{\prime\prime}y^{\prime 2}+f_{y}^{\prime}y^{\prime\prime},$
 $y^{(4)}\;=\;f_{xxx}^{\prime\prime\prime}+3f_{xxy}^{\prime\prime\prime}y^{% \prime}+3f_{xyy}^{\prime\prime\prime}y^{\prime 2}+f_{yyy}^{\prime\prime\prime}% y^{\prime 3}+3f_{xy}^{\prime\prime}y^{\prime\prime}+3f_{yy}^{\prime\prime}y^{% \prime}y^{\prime\prime}+f_{y}^{\prime}y^{\prime\prime\prime}$

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

 $\displaystyle f_{x}^{\prime}(x,\,y)+f_{y}^{\prime}(x,\,y)f(x,\,y)\;=\;0$ (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 DerivativesOfSolutionOfFirstOrderODE 2013-03-22 18:59:14 2013-03-22 18:59:14 pahio (2872) pahio (2872) 10 pahio (2872) Theorem msc 34A12 msc 34-00 SolutionsOfOrdinaryDifferentialEquation InflexionPoint