The differential equation `(dy)/(dx)=sqrt(1-y^(2))/(y)` determines a family of circles with

Options

A. Variable radil and fixed centre at `(0,1)`
B. Variable radii and fixed centre at `(0, -1)`
C. Fixed radius 1 and variable centres along y-axis
D. Fixed radius 1 and variable centres along x-axis

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{y}\), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to separate the variables \(y\) and \(x\):
\[ y \, dy = \sqrt{1 - y^2} \, dx \]

### Step 2: Integrating Both Sides
Next, we will integrate both sides. The left side involves \(y\) and the right side involves \(x\):
\[ \int y \, dy = \int \sqrt{1 - y^2} \, dx \]

### Step 3: Substituting for Integration
To simplify the integration, we can use a substitution. Let \(u = 1 - y^2\). Then, we have:
\[ du = -2y \, dy \quad \Rightarrow \quad y \, dy = -\frac{1}{2} du \]
Now, substituting this into our integral gives:
\[ -\frac{1}{2} \int du = \int \sqrt{u} \, dx \]

### Step 4: Integrating
Now we integrate both sides:
\[ -\frac{1}{2} u = \frac{2}{3} x + C \]
Substituting back \(u = 1 - y^2\):
\[ -\frac{1}{2} (1 - y^2) = \frac{2}{3} x + C \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ 1 - y^2 = -\frac{4}{3} x - 2C \]
Let \(C' = -2C\), we can rewrite it as:
\[ y^2 = 1 + \frac{4}{3} x + C' \]

### Step 6: Recognizing the Circle Equation
This can be rearranged to:
\[ y^2 + \left( x + \frac{2}{3} \right)^2 = 1 + C' \]
This represents a family of circles with:
- Center at \(\left(-\frac{2}{3}, 0\right)\)
- Radius \(\sqrt{1 + C'}\)

### Conclusion
The correct interpretation of the differential equation is that it describes a family of circles with a fixed radius and variable centers along the x-axis.