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

Options

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

Answer

To solve the given differential equation and find \( f(-\frac{1}{2}) \), we will follow these steps:

### Step 1: Write the given differential equation
The differential equation given is:
\[ y(1 + xy)dx = xdy \]

### Step 2: Rearrange the equation
We can rearrange the equation by dividing both sides by \( x \):
\[ \frac{y(1 + xy)}{x}dx = dy \]

### Step 3: Express in terms of \( \frac{dy}{dx} \)
This can be rewritten as:
\[ \frac{y(1 + xy)}{x} = \frac{dy}{dx} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{y(1 + xy)}{x} \]

### Step 4: Substitute \( y = vx \)
Let \( y = vx \), where \( v \) is a function of \( x \). Then:
\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]
Substituting this into the equation gives:
\[ v + x\frac{dv}{dx} = \frac{vx(1 + vx)}{x} \]
This simplifies to:
\[ v + x\frac{dv}{dx} = v(1 + vx) \]

### Step 5: Simplify the equation
Rearranging gives:
\[ x\frac{dv}{dx} = v^2x \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ \frac{dv}{dx} = v^2 \]

### Step 6: Separate variables and integrate
Separating variables, we have:
\[ \frac{dv}{v^2} = dx \]
Integrating both sides:
\[ -\frac{1}{v} = x + C \]
Thus:
\[ v = -\frac{1}{x + C} \]

### Step 7: Substitute back for \( y \)
Since \( v = \frac{y}{x} \), we have:
\[ y = -\frac{x}{x + C} \]

### Step 8: Use the point (1, -1) to find \( C \)
Substituting the point \( (1, -1) \):
\[ -1 = -\frac{1}{1 + C} \]
This implies:
\[ 1 + C = 1 \implies C = 0 \]
Thus, the equation becomes:
\[ y = -\frac{x}{x} = -1 \]

### Step 9: Find \( f(-\frac{1}{2}) \)
Now we need to find \( f(-\frac{1}{2}) \):
\[ f(-\frac{1}{2}) = -\frac{-\frac{1}{2}}{-\frac{1}{2}} = -1 \]

### Final Result
Thus, the value of \( f(-\frac{1}{2}) \) is:
\[ \boxed{-1} \]