The singular solution of the differential equation $y=px+p-p^{3}$ is given by :

Question

Accepted Answer

To determine the singular solution of the differential equation given by $ y = px + p - p^3 $, we need to analyze the structure of the equation and the role of the parameter $ p $.

### Explanation for Option 2:

The given differential equation is $ y = px + p - p^3 $. This is a parametric representation where $ p $ is a parameter. To find the singular solution, we need to eliminate the parameter $ p $ from the equation.

1. **Differentiate with respect to $ x $:**
   $$
   \frac{dy}{dx} = p + \frac{dp}{dx}(x + 1 - 3p^2)
   $$
   For the singular solution, the derivative $ \frac{dy}{dx} $ should be independent of $ p $. This implies:
   $$
   x + 1 - 3p^2 = 0 \quad \Rightarrow \quad x = 3p^2 - 1
   $$

2. **Substitute $ x = 3p^2 - 1 $ back into the original equation:**
   $$
   y = p(3p^2 - 1) + p - p^3 = 3p^3 - p + p - p^3 = 2p^3
   $$

3. **Express $ p $ in terms of $ x $:**
   $$
   x = 3p^2 - 1 \quad \Rightarrow \quad p^2 = \frac{x + 1}{3} \quad \Rightarrow \quad p = \sqrt{\frac{x + 1}{3}}
   $$

4. **Substitute $ p $ back into the expression for $ y $:**
   $$
   y = 2p^3 = 2\left(\sqrt{\frac{x + 1}{3}}\right)^3 = 2\left(\frac{x + 1}{3}\right)^{3/2}
   $$

Thus, the singular solution is $ y = 2\left(\frac{x + 1}{3}\right)^{3/2} $, which corresponds to Option 2.

### Explanation for why other options are incorrect:

- **Option 1: $ y = \frac{1}{2}\left(\frac{x+1}{3}\right)^{3/2} $**
  - This option incorrectly scales the singular solution by a factor of $ \frac{1}{2} $ instead of 2. The derivation clearly shows that the correct factor is 2.

- **Option 3: $ y = \frac{1}{2}\left(\frac{x-1}{3}\right)^{3/2} $**
  - This option not only has the incorrect scaling factor but also uses $ x-1 $ instead of $ x+1 $. The correct expression derived from the elimination of $ p $ involves $ x+1 $.

- **Option 4: $ y = 2\left(\frac{x-1}{3}\right)^{3/2} $**
  - While this option has the correct scaling factor of 2, it incorrectly uses $ x-1 $ instead of $ x+1 $. The derivation shows that the correct expression involves $ x+1 $.

In conclusion, Option 2 is the correct singular solution as it accurately reflects the derivation process and the necessary conditions for eliminating the parameter $ p $.