Question

The solution of \(y = xp + \frac{m}{p}\) where \(p = \frac{dy}{dx}\) is

• A. \(y = \frac{m}{c}\)
• B. \(y = xc\)
• C. \(y = xc - \frac{m}{c}\)
• D. \(y = xc + \frac{m}{c}\)

Hint:

For any equation in the form of \(y = xp + f(p)\) (Clairaut's equation), the general solution is always found by simply replacing the parameter \(p\) with a constant \(c\). This is a valuable shortcut in exams.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The given differential equation \(y = xp + \frac{m}{p}\) is a classic example of Clairaut's equation, which has the general form \(y = xp + f(p)\), where \(p = \frac{dy}{dx}\). Clairaut's equations have a general solution and a singular solution. The general solution is found by simply replacing \(p\) with an arbitrary constant \(c\).
Step 2: Key Formula or Approach:
To solve a Clairaut's equation \(y = xp + f(p)\):
1. Differentiate the entire equation with respect to \(x\).
2. Use the fact that \(\frac{dy}{dx} = p\).
3. The resulting equation can be factored. One factor will lead to the general solution, and the other to the singular solution.
Step 3: Detailed Explanation:
Given the equation: \[ y = xp + \frac{m}{p} \quad (*).\] Differentiate with respect to \(x\): \[ \frac{dy}{dx} = \left(1 \cdot p + x \cdot \frac{dp}{dx}\right) - \frac{m}{p^2} \frac{dp}{dx} \] Since \(\frac{dy}{dx} = p\), we have: \[ p = p + x \frac{dp}{dx} - \frac{m}{p^2} \frac{dp}{dx} \] Subtract \(p\) from both sides: \[ 0 = x \frac{dp}{dx} - \frac{m}{p^2} \frac{dp}{dx} \] Factor out \(\frac{dp}{dx}\): \[ \left(x - \frac{m}{p^2}\right) \frac{dp}{dx} = 0 \] This equation gives two possibilities:
Case 1: \(\frac{dp}{dx} = 0\)
If \(\frac{dp}{dx} = 0\), integrating with respect to \(x\) gives \(p = c\), where \(c\) is an arbitrary constant.
Substituting \(p=c\) back into the original equation (*), we get the general solution: \[ y = xc + \frac{m}{c} \] This matches option (4).
Case 2: \(x - \frac{m}{p^2} = 0\)
This leads to \(p^2 = \frac{m}{x}\) or \(p = \pm\sqrt{\frac{m}{x}}\). Substituting this back into the original equation gives the singular solution, which is not among the options.
Step 4: Final Answer:
The general solution of the Clairaut's equation is obtained by replacing \(p\) with a constant \(c\), which gives \(y = xc + \frac{m}{c}\).