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.