Question

\(y = mx + \dfrac{2}{m}\) is the general solution of

• A. \( y\left(\dfrac{dy}{dx}\right)^2 = x\left(\dfrac{dy}{dx}\right) + 2 \)
• B. \( y = x\dfrac{dy}{dx} + 2 \)
• C. \( y\left(\dfrac{dy}{dx}\right) = x\left(\dfrac{dy}{dx}\right)^2 + 2 \)
• D. \( y\left(\dfrac{dy}{dx}\right) = x + 2 \)

Hint:

To find the differential equation of a family of curves, differentiate and eliminate the arbitrary constant or parameter.

Answer 1

The Correct Option is C

Step 1: Differentiate the given solution.
Given \[ y = mx + \frac{2}{m} \] Differentiate w.r.t. \(x\): \[ \frac{dy}{dx} = m \] Step 2: Eliminate the parameter \(m\).
From above, \[ m = \frac{dy}{dx} \] Step 3: Substitute in the original equation.
\[ y = x\frac{dy}{dx} + \frac{2}{\frac{dy}{dx}} \] Step 4: Simplify.
Multiplying both sides by \(\frac{dy}{dx}\): \[ y\left(\frac{dy}{dx}\right) = x\left(\frac{dy}{dx}\right)^2 + 2 \] Step 5: Final conclusion.
Hence, the required differential equation is option (C).