Question

The differential equation of all non-horizontal lines in a plane is __________________.

Answer 1

Equation of all non-horizontal lines in a plane is ax + by + c = 0, where a ≠ 0.

$$\begin{gathered} ax+by+c=0 \\ \mathrm{Differentiating}\mathrm{both}\mathrm{the}\mathrm{sides}\mathrm{w}.\mathrm{r}.\mathrm{t}.y \\ \Rightarrow a\frac{dx}{dy}+b=0 \\ \mathrm{Again}\mathrm{differentiating}\mathrm{both}\mathrm{the}\mathrm{sides}\mathrm{w}.\mathrm{r}.\mathrm{t}.y \\ \Rightarrow a\frac{d^{2}x}{d{y}^2}=0 \\ \Rightarrow \frac{d^{2}x}{d{y}^2}=0\left(\because a\ne 0\right) \end{gathered}$$


Hence, the differential equation of all non-horizontal lines in a plane is $\underline{\frac{d^{2}x}{d{y}^2}=0}.$