The solution of the differential equation `(dy)/(dx)+(xy)/(1-x^(2))=xsqrty,(|x| lt 1)` is `sqrty=-(f(x))/(3)+C(1-x^(2))^((1)/(4))`, where `f((1)/(2))=(3)/(4) and C` is an arbitrary constant. Then, the value of `f(-(1)/(2))` is

Options

A. `-(3)/(4)`
B. `(3)/(4)`
C. `(1)/(4)`
D. `(3)/(2)`

Answer

To solve the given problem step by step, we will start with the differential equation and manipulate it to find the required function \( f(x) \).

### Step 1: Rewrite the Differential Equation
The given differential equation is:

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

### Step 2: Substitute \( \sqrt{y} = t \)
Let \( \sqrt{y} = t \). Then, differentiating both sides gives:

\[ \frac{1}{2\sqrt{y}} \frac{dy}{dx} = \frac{dt}{dx} \]

This implies:

\[ \frac{dy}{dx} = 2t \frac{dt}{dx} \]

### Step 3: Substitute into the Differential Equation
Substituting into the original equation gives:

\[ 2t \frac{dt}{dx} + \frac{xt^2}{1 - x^2} = xt \]

### Step 4: Rearranging the Equation
Rearranging the equation yields:

\[ 2t \frac{dt}{dx} = xt - \frac{xt^2}{1 - x^2} \]

### Step 5: Factor Out \( x \)
Factoring out \( x \) from the right side gives:

\[ 2t \frac{dt}{dx} = x\left( t - \frac{t^2}{1 - x^2} \right) \]

### Step 6: Simplifying
This can be simplified to:

\[ 2t \frac{dt}{dx} = \frac{xt(1 - x^2 - t)}{1 - x^2} \]

### Step 7: Separate Variables
Separate the variables:

\[ \frac{2t}{t(1 - x^2 - t)} dt = \frac{x}{1 - x^2} dx \]

### Step 8: Integrate Both Sides
Integrate both sides:

\[ \int \frac{2}{1 - x^2 - t} dt = \int \frac{x}{1 - x^2} dx \]

### Step 9: Solve the Integrals
The left side will yield a logarithmic function, and the right side will yield a negative logarithmic function. After integration, we can express the solution in terms of \( t \) and \( x \).

### Step 10: Express \( \sqrt{y} \)
After integration, we can express \( \sqrt{y} \) in terms of \( x \) and an arbitrary constant \( C \):

\[ \sqrt{y} = -\frac{f(x)}{3} + C(1 - x^2)^{\frac{1}{4}} \]

### Step 11: Find \( f(x) \)
From the problem, we know that \( f\left(\frac{1}{2}\right) = \frac{3}{4} \). We need to find \( f\left(-\frac{1}{2}\right) \).

### Step 12: Substitute \( x = -\frac{1}{2} \)
Substituting \( x = -\frac{1}{2} \) into the expression for \( f(x) \):

\[ f\left(-\frac{1}{2}\right) = 1 - \left(-\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4} \]

### Conclusion
Thus, the value of \( f\left(-\frac{1}{2}\right) \) is:

\[ \boxed{\frac{3}{4}} \]