Question

The differential equation of all non-vertical lines in a plane is:
A. \[\dfrac{{{d^2}y}}{{d{x^2}}} = 0\]
B. $\dfrac{{{d^2}x}}{{d{y^2}}} = 0$
C. $\dfrac{{dy}}{{dx}} = 0$
D. $\dfrac{{dx}}{{dy}} = 0$

Answer 1

Hint: First we write the general equation of non-vertical lines in a plane and then differentiate the equation two times with respect to y and after that we get our answer of The differential equation of all non-vertical lines in a plane. We can also calculate it with a different method.

Complete step-by-step solution
The general equation of non-vertical lines in a plane is
$
  ax + by + c = 0 \\
  {\text{where}} \\
  b \ne 0 \\
 $
On differentiating the equation with respect to y we get
$
  a + b\dfrac{{dy}}{{dx}} = 0 \\
$
Again differentiating the equation with respect to y, we get
$
  {\text{b}}\dfrac{{{d^2}y}}{{d{x^2}}} = 0 \\
   \Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} = 0 \\
 $
Since we get the right answer
Hence option A is the correct answer.

Note: First write the general equation of non-vertical lines in a plane which is $ax + by + c = 0$ where. We have to remember this formula. Then differentiated it with respect to $y$ and we get $a + b\dfrac{{dy}}{{dx}} = 0$ and by differentiating again we get $\dfrac{{d{y^2}}}{{d{x^2}}} = 0$ we get our answer. We can also solve this problem by using the equation $y = mx + c$ so and differentiate the equation two times with respect to $y$ we will get the same correct answer.