Question

\( y = c^2 + \dfrac{c}{x} \) is the solution of the differential equation

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

Hint:

To eliminate constants, express them in terms of derivatives and substitute back into the original equation.

Answer 1

The Correct Option is B

Step 1: Differentiate the given solution.
\[ y = c^2 + \frac{c}{x} \] \[ \frac{dy}{dx} = -\frac{c}{x^2} \] Step 2: Express \( c \) in terms of \( x \) and \( \frac{dy}{dx} \).
\[ c = -x^2 \frac{dy}{dx} \] Step 3: Substitute into the original equation.
\[ y = c^2 + \frac{c}{x} \] \[ y = x^4\left(\frac{dy}{dx}\right)^2 - x\left(\frac{dy}{dx}\right) \] Step 4: Rearrange the equation.
\[ x^4\left(\frac{dy}{dx}\right)^2 - x\left(\frac{dy}{dx}\right) - y = 0 \] Step 5: Conclusion.
Hence, the required differential equation is \[ x^4\left(\frac{dy}{dx}\right)^2 - x\left(\frac{dy}{dx}\right) - y = 0 \]