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 number of solution of the equation `f(x)=-1` and the singular solution of the equation `y=x(dy)/(dx)+((dy)/(dx))^(2)` is

Options

A. 1
B. 2
C. 4
D. 0

Explanation

To solve the given differential equation \( y = px + f(p) \), where \( p = \frac{dy}{dx} \), we will follow the steps outlined in the video transcript.

### Step 1: Write the given equation
We start with the equation:
\[ y = px + f(p) \tag{1} \]
where \( p = \frac{dy}{dx} \).

### Step 2: Differentiate with respect to \( x \)
Next, we differentiate equation (1) with respect to \( x \):
\[ \frac{dy}{dx} = p + x \frac{dp}{dx} + \frac{df}{dp} \cdot \frac{dp}{dx} \]
Since \( p = \frac{dy}{dx} \), we can substitute \( p \) into the equation:
\[ p = p + x \frac{dp}{dx} + f'(p) \frac{dp}{dx} \]
This simplifies to:
\[ 0 = x \frac{dp}{dx} + f'(p) \frac{dp}{dx} \tag{2} \]

### Step 3: Factor out \( \frac{dp}{dx} \)
We can factor out \( \frac{dp}{dx} \):
\[ \frac{dp}{dx} (x + f'(p)) = 0 \]
This gives us two cases:
1. \( \frac{dp}{dx} = 0 \) which implies \( p = c \) (a constant).
2. \( x + f'(p) = 0 \) which leads to \( f'(p) = -x \).

### Step 4: Analyze the first case
For the first case \( p = c \):
Substituting \( p = c \) back into equation (1):
\[ y = cx + f(c) \]
This represents a family of straight lines.

### Step 5: Analyze the second case
For the second case \( f'(p) = -x \):
We need to eliminate \( p \) from equation (1). From \( f'(p) = -x \), we can express \( x \) in terms of \( p \):
\[ x = -f'(p) \]
Substituting this back into equation (1):
\[ y = p(-f'(p)) + f(p) \]
This gives us a new equation in terms of \( p \).

### Step 6: Solve for \( f(x) = -1 \)
Now we need to find the number of solutions for the equation \( f(x) = -1 \). From our earlier work, we can substitute and analyze the resulting equation to find the number of solutions.

### Step 7: Find singular solution
To find the singular solution of the equation \( y = x \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 \):
We can substitute \( p = \frac{dy}{dx} \) into the equation and analyze the resulting expression to find the singular solution.

### Conclusion
The number of solutions for \( f(x) = -1 \) is determined by the roots of the resulting equation, and the singular solution can be derived from the conditions we have set up.