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 $f_x'(x,\,y)$ and $f_y'(x,\,y)$ .

If $y(x)$ is a solution of the first order ordinary differential equation

(1)

(2)

(3)

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 $n\!+\!1$ .

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

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

(4)

Bibliography

1

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