If the curve `y=f(x)` passes through the point (1, -1) and satisfies the differential equation :
`y(1+xy)dx=x dy`, then `f(- 1/2)` is equal to :

Options

A. `- 4/5`
B. `2/5`
C. `4/5`
D. `- 2/5`

Answer

To solve the problem, we need to find the function \( y = f(x) \) that satisfies the given differential equation and passes through the point (1, -1). The differential equation is:

\[ y(1 + xy)dx = x dy \]

### Step 1: Rearranging the Differential Equation
We start by rearranging the given differential equation. We can rewrite it as:

\[ y(1 + xy)dx - x dy = 0 \]

### Step 2: Dividing by \( y^2 \)
Next, we divide the entire equation by \( y^2 \):

\[ \frac{y(1 + xy)}{y^2}dx - \frac{x}{y^2}dy = 0 \]

This simplifies to:

\[ \frac{1 + xy}{y}dx - \frac{x}{y^2}dy = 0 \]

### Step 3: Rearranging the Terms
Rearranging gives us:

\[ \frac{1 + xy}{y}dx = \frac{x}{y^2}dy \]

### Step 4: Expressing in Terms of Derivatives
Now we can express this in terms of derivatives:

\[ \frac{dx}{dy} = \frac{xy^2}{1 + xy} \]

### Step 5: Separating Variables
Next, we separate the variables:

\[ \frac{1 + xy}{x}dx = \frac{y^2}{y}dy \]

This gives:

\[ \frac{1 + xy}{x}dx = y dy \]

### Step 6: Integrating Both Sides
Now we integrate both sides. The left side can be integrated as:

\[ \int \left( \frac{1}{x} + y \right) dx = \int y dy \]

This results in:

\[ \ln |x| + \frac{y^2}{2} = C \]

### Step 7: Solving for \( y \)
We can solve for \( y \):

\[ y^2 = 2C - 2\ln |x| \]

### Step 8: Finding the Constant \( C \)
To find the constant \( C \), we use the initial condition that the curve passes through the point (1, -1):

\[ (-1)^2 = 2C - 2\ln |1| \]

This simplifies to:

\[ 1 = 2C \implies C = \frac{1}{2} \]

### Step 9: Substituting \( C \) Back
Now we substitute \( C \) back into the equation for \( y \):

\[ y^2 = 1 - \ln |x| \]

### Step 10: Finding \( f(-\frac{1}{2}) \)
Now we need to find \( f(-\frac{1}{2}) \):

\[ y^2 = 1 - \ln \left| -\frac{1}{2} \right| = 1 - \ln \frac{1}{2} = 1 + \ln 2 \]

Thus,

\[ y = \pm \sqrt{1 + \ln 2} \]

Since we are looking for \( f(-\frac{1}{2}) \), we take the negative value (as the curve passes through (1, -1)):

\[ f(-\frac{1}{2}) = -\sqrt{1 + \ln 2} \]

### Final Answer
The value of \( f(-\frac{1}{2}) \) is:

\[ f(-\frac{1}{2}) = -\sqrt{1 + \ln 2} \]