derivatives of solution of first order ODE

Suppose that $f$ is a continuously differentiable function defined on an open subset $E$ of ${ℝ}^2$, 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

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

$$y^{\prime \prime \prime }=f_{xx}^{\prime \prime }+2f_{xy}^{\prime \prime }{y}^{\prime }+f_{yy}^{\prime \prime }{y}^{\prime 2}+f_y^{\prime }{y}^{\prime \prime },$$

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

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

$f_x^{\prime }(x,y)+f_y^{\prime }(x,y)f(x,y)=\mathrm{\hspace{0.33em}0}$ (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.

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