Suppose y is a function of x that satisfies `(dy)/(dx)=(sqrt(1-y^(2)))/(x^(2))` and y=0 at `x=(2)/(pi)` then `y((3)/(pi))` is equal to :

Options

A. 0
B. `(1)/(2)`
C. 1
D. 2

Explanation

To solve the given differential equation and find the value of \( y \) at \( x = \frac{3}{\pi} \), we will follow these steps:

### Step 1: Write the differential equation
The given differential equation is:
\[ \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{x^2} \]

### Step 2: Separate the variables
We can separate the variables by rewriting the equation as:
\[ \frac{dy}{\sqrt{1 - y^2}} = \frac{dx}{x^2} \]

### Step 3: Integrate both sides
Now, we will integrate both sides:
\[ \int \frac{dy}{\sqrt{1 - y^2}} = \int \frac{dx}{x^2} \]
The left side integrates to:
\[ \sin^{-1}(y) \]
The right side integrates to:
\[ -\frac{1}{x} + C \]
Thus, we have:
\[ \sin^{-1}(y) = -\frac{1}{x} + C \]

### Step 4: Apply the initial condition
We know that \( y = 0 \) when \( x = \frac{2}{\pi} \). Plugging these values into the equation gives:
\[ \sin^{-1}(0) = -\frac{1}{\frac{2}{\pi}} + C \]
This simplifies to:
\[ 0 = -\frac{\pi}{2} + C \]
Thus, we find:
\[ C = \frac{\pi}{2} \]

### Step 5: Write the equation of the curve
Substituting \( C \) back into the equation gives:
\[ \sin^{-1}(y) = -\frac{1}{x} + \frac{\pi}{2} \]

### Step 6: Find \( y \) at \( x = \frac{3}{\pi} \)
Now we need to find \( y \) when \( x = \frac{3}{\pi} \):
\[ \sin^{-1}(y) = -\frac{1}{\frac{3}{\pi}} + \frac{\pi}{2} \]
This simplifies to:
\[ \sin^{-1}(y) = -\frac{\pi}{3} + \frac{\pi}{2} \]
Finding a common denominator (6):
\[ \sin^{-1}(y) = -\frac{2\pi}{6} + \frac{3\pi}{6} = \frac{\pi}{6} \]

### Step 7: Solve for \( y \)
Taking the sine of both sides:
\[ y = \sin\left(\frac{\pi}{6}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Final Answer
Thus, the value of \( y \) at \( x = \frac{3}{\pi} \) is:
\[ y\left(\frac{3}{\pi}\right) = \frac{1}{2} \]