Question

General solution of \[ xp^2-yp+a=0, \] is \( \_\_\_\_ \), where \[ p=\frac{dy}{dx}. \]

• A. \(y=cx+\dfrac{a}{c}\)
• B. \(y=cx-\dfrac{a}{c^2}\)
• C. \(y=cx+\dfrac{c^2}{a}\)
• D. \(y=x^2+\dfrac{a}{c^2}\)

Hint:

For Clairaut's equation \(y=px+f(p)\), the general solution is obtained by replacing \(p\) with constant \(c\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The given equation is a first-order, higher-degree differential equation. By rearranging it, we can check if it fits the form of Clairaut's equation, which is \(y = px + f(p)\).

Step 2: Key Formula or Approach:
1. Rearrange the given differential equation to solve for \(y\). 2. If the equation is in the form \(y = px + f(p)\), it is Clairaut's equation. 3. The general solution of Clairaut's equation is obtained by replacing the parameter \(p\) with an arbitrary constant \(c\).

Step 3: Detailed Explanation:
The given differential equation is: \[ xp^2 - yp + a = 0 \] We want to solve for \(y\). Rearrange the terms: \[ yp = xp^2 + a \] Assuming \(p \neq 0\), we can divide by \(p\): \[ y = \frac{xp^2}{p} + \frac{a}{p} \] \[ y = xp + \frac{a}{p} \] This equation is now in the standard form of Clairaut's equation, \(y = px + f(p)\), where \(f(p) = \frac{a}{p}\).
To find the general solution, we replace every instance of \(p\) with an arbitrary constant, which we'll call \(c\). \[ y = cx + \frac{a}{c} \] This gives the general solution, which represents a family of straight lines.

Step 4: Final Answer:
The general solution of the given Clairaut's equation is \(y = cx + \frac{a}{c}\), which corresponds to option (A).