Question

The differential equation for the line $y=mx+c$ is
A. $\dfrac{\text{d}y}{\text{d}x}=m$
B. $\dfrac{\text{d}y}{\text{d}x}+m=0$
C. $\dfrac{\text{d}y}{\text{d}x}=0$
D. None of these.

Answer 1

Hint: Differentiating the straight line equation with respect to x will give its differential form where the differentiation of constant will be 0. Further, rearrange the equation in its differential equation form.

Formula used: $\dfrac{\text{d}}{\text{d}x}\lgroup~x\rgroup=1$

Complete step by step solution: The given equation is the equation of the straight line written as follows,
$y=mx+c$.........(i)
To get the required differential equation we need to differentiate the equation as mentioned above with respect to x.
This gives,
$\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}}{\text{d}x}\lgroup~mx+c\rgroup$.........(ii)
We know,
$\dfrac{\text{d}}{\text{d}x}\lgroup~x\rgroup=1$
So, equation (ii) implies,
$\dfrac{\text{d}y}{\text{d}x}=m$ [Differentiation of the constant is always 0.]
Thus, the above equation is the differential equation for the given line $y=mx+c$

Thus, Option (A) is correct.

Additional information: The equation $y=mx+c$ is one of the general forms of the equation of a straight line. Here, m is the slope and c is the intercept on the y-axis.

Note: It is noted that c is the intercept in the equation $y=mx+c$. So, here c will be a constant term. Many do the mistake of differentiating c. Here, the differentiation of c will be 0.