Question

The differential equation of all family of lines $y = mx + \frac{4}{m}$ obtained by eliminating the arbitrary constant $m$ is

• A. $y \left(\frac{dy}{dx}\right) = 4$
• B. $x \left(\frac{dy}{dx}\right)^2 + y \left(\frac{dy}{dx}\right) + 4 = 0$
• C. $x \left(\frac{dy}{dx}\right) + 4 = 0$
• D. $x \left(\frac{dy}{dx}\right)^2 - y \left(\frac{dy}{dx}\right) + 4 = 0$

Hint:

The given equation $y = mx + \frac{4}{m}$ is a classic example of Clairaut's form of a differential equation, which generally looks like $y = x\left(\frac{dy}{dx}\right) + f\left(\frac{dy}{dx}\right)$. For any such equation, you can substitute $m$ with $\frac{dy}{dx}$ directly to obtain the differential equation in a single step!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a family of straight lines containing an arbitrary parameter $m$. The objective is to form the corresponding differential equation by eliminating this parameter $m$.

Step 2: Key Formula or Approach:
1. Differentiate the given equation with respect to $x$ to find an expression for the slope $m$ in terms of $\frac{dy}{dx}$. 2. Substitute the value of $m$ back into the original family equation to eliminate the parameter completely.

Step 3: Detailed Explanation:
The given equation of the family of lines is: $$y = mx + \frac{4}{m} \quad \text{--- (Equation 1)}$$ Differentiating both sides with respect to $x$: $$\frac{dy}{dx} = m(1) + 0 \implies m = \frac{dy}{dx}$$ Now substitute this value of $m$ back into Equation 1: $$y = \left(\frac{dy}{dx}\right)x + \frac{4}{\left(\frac{dy}{dx}\right)}$$ To clear the fraction, multiply the entire equation by $\frac{dy}{dx}$: $$y \left(\frac{dy}{dx}\right) = x \left(\frac{dy}{dx}\right)^2 + 4$$ Rearranging all terms to one side of the equation yields: $$x \left(\frac{dy}{dx}\right)^2 - y \left(\frac{dy}{dx}\right) + 4 = 0$$

Step 4: Final Answer:
The resulting differential equation is $x \left(\frac{dy}{dx}\right)^2 - y \left(\frac{dy}{dx}\right) + 4 = 0$, which corresponds to option (D).