Question

Obtain the differential equation representing family of lines y=mx+c (m and c are arbitrary constant).

Answer 1

Answer :

The given family of lines is

y = mx + c, where m and c are arbitrary constant

Now, differentiating both sides with respect to x, we get

      $\frac{dy}{dx}=m$

Again, differentiating both sides with respect to x, we get

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

which is the required differential equation.

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO DETERMINE

The differential equation representing family of lines y=mx + c (m and c are arbitrary constant)

CALCULATION

The given equation of family of lines is

$\sf{y \: = mx + c}$

Differentiating both sides with respect to x we get

$\displaystyle \sf{ \frac{dy}{dx} \: = m \: }$

Again Differentiating both sides with respect to x we get

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

Hence the required differential equation is

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

# POINT TO NOTICE

1. The order of the differential equation is 2

2. Since the given equation of family of lines are consisting of 2 arbitrary constants

So the order of the differential equation is 2

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