Find the differential equation of all lines in a 2-D plane.
My Attempt
For horizontal lines,we have $$\frac{dy}{dx}=0$$
For vertical lines,we have $$\frac{dx}{dy}=0$$
For non-vertical lines(includes the case when line is horizontal),we have $$\frac{d^2y}{dx^2}=0$$
For non-horizontal lines(includes the case when it is vertical),we have $$\frac{d^2x}{dy^2}=0$$
Similarly,can we have a single differential equation in which all the above cases are accounted for
Best Answer
You can neither express all lines in terms of a function $y(x)$ nor express all lines in terms of a function $x(y)$, since in each case there are lines that aren't described by such functions. This is a matter of the form in which you represent the functions, not of the differential equations that you impose on that form.
All lines can be represented in the form $(x(s),y(s))$, where $s$ is an arc length parameter, and in this representation the differential equation for lines is $\frac{\mathrm d^2}{\mathrm ds^2}(x(s),y(s))=0$.