The differential equation `y=px+f(p)`, …………..(i)
where `p=(dy)/(dx)`,is known as Clairout's equation. To solve equation i) differentiate it with respect to x, which gives either
`(dp)/(dx)=0 rArr p =c`………….(ii)
or `x+f^(i)(p)=0`…………(iii)
The singular solution of the differential equation `y=mx + m-m^(3)`, where `m=(dy)/(dx)`, passes through the point.

Options

A. (0,0)
B. (0,1)
C. (1,0)
D. (-1,0)

Explanation

To solve the given differential equation \( y = mx + m - m^3 \), where \( m = \frac{dy}{dx} \), we will follow the steps outlined in Clairaut's equation.

### Step 1: Differentiate the equation
We start with the equation:
\[ y = mx + m - m^3 \]
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = m + x \frac{dm}{dx} + \frac{dm}{dx} - 3m^2 \frac{dm}{dx} \]
This simplifies to:
\[ \frac{dy}{dx} = m + (x + 1 - 3m^2) \frac{dm}{dx} \]

### Step 2: Substitute \( \frac{dy}{dx} \)
Since \( \frac{dy}{dx} = m \), we substitute this into the equation:
\[ m = m + (x + 1 - 3m^2) \frac{dm}{dx} \]
Subtracting \( m \) from both sides gives:
\[ 0 = (x + 1 - 3m^2) \frac{dm}{dx} \]

### Step 3: Analyze the equation
This equation implies two cases:
1. \( \frac{dm}{dx} = 0 \) which means \( m \) is a constant.
2. \( x + 1 - 3m^2 = 0 \) which leads to a relationship between \( x \) and \( m \).

### Step 4: Solve for constant \( m \)
If \( \frac{dm}{dx} = 0 \), then \( m = c \) (a constant). The corresponding equation becomes:
\[ y = cx + c - c^3 \]

### Step 5: Solve for \( m \) in terms of \( x \)
From the second case, we set:
\[ x + 1 - 3m^2 = 0 \implies 3m^2 = x + 1 \implies m^2 = \frac{x + 1}{3} \]
Thus, we have:
\[ m = \pm \sqrt{\frac{x + 1}{3}} \]

### Step 6: Substitute \( m \) back into the original equation
Substituting \( m \) back into the original equation \( y = mx + m - m^3 \):
\[ y = \sqrt{\frac{x + 1}{3}} x + \sqrt{\frac{x + 1}{3}} - \left(\sqrt{\frac{x + 1}{3}}\right)^3 \]
Calculating \( m^3 \):
\[ m^3 = \left(\frac{x + 1}{3}\right)^{3/2} \]
Thus,
\[ y = \sqrt{\frac{x + 1}{3}} x + \sqrt{\frac{x + 1}{3}} - \frac{(x + 1)^{3/2}}{3\sqrt{3}} \]

### Step 7: Find the singular solution
To find the singular solution, we need to check the point where \( y = 0 \):
Setting \( y = 0 \) and solving for \( x \):
\[ 0 = \sqrt{\frac{x + 1}{3}} x + \sqrt{\frac{x + 1}{3}} - \frac{(x + 1)^{3/2}}{3\sqrt{3}} \]
This will lead us to find the specific point.

### Conclusion
After solving for \( x \) when \( y = 0 \), we find that the singular solution passes through the point \( (-1, 0) \).