Question

The differential equation of all non horizontal lines in a plane is ?​

Answer 1

Answer:

$\frac{d^2x}{dy^2}=0$

Step-by-step explanation:-

Let the  family of all non-horizontal lines in xy-plane be,

y = mx + c

Differentiating with respect to y (Why?)

$1=m\frac{dx}{dy}+0\\\;\\\frac{dx}{dy}=\frac{1}{m}$

Again differentiating this equation with respect to y

$\frac{d^2x}{dy^2}=0$

This is the required differential equation of all non-horizontal lines in xy-plane.

Note:-Why we have differentiated with respect to y ? Here are some reasons:-

1) Since the family of all non horizontal line contains vertical lines also. Slope of a vertical line dy/dx is ∞ and we won't be able to proceed further. In other words dx will be zero and won't be defined.

2)If we differentiate with respect to x, the differential equation will be,

d²y/dx² = 0

Integrating the above equation,

dy/dx = c

dy = c.dx

Again Integrating the above equation

y = cx

Here if c = 0, ⇒ y = 0

Which is equation of a horizontal line. Hence there is contradiction.

Answer 2

$\huge{\underline{\mathtt{\red{A}\pink{N}\green{S}\blue{W}\purple{E}\orange{R}}}}$

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

$\frac{ {d}^{2}x }{d {y}^{2} } = 0$