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

Options

A. variable radii and a fixed centre at (0, 1)
B. variable radii and a fixed centre at (0, –1)
C. fixed radius 1 and variable centres along the X-axis
D. fixed radius 1 and variable centres along the Y-axis

Answer

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

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

### Step 2: Integrate Both Sides
Next, we integrate both sides. The left side requires a substitution. Let:
\[ t = 1 - y^2 \quad \Rightarrow \quad dt = -2y \, dy \quad \Rightarrow \quad dy = \frac{dt}{-2y} \]
Substituting this into the integral gives:
\[ \int \frac{y}{\sqrt{t}} \left(-\frac{dt}{2y}\right) = \int -\frac{1}{2\sqrt{t}} \, dt \]
Thus, we have:
\[ -\frac{1}{2} \int t^{-1/2} \, dt = -\sqrt{t} + C \]
Now, integrating the right side:
\[ \int dx = x + C \]

### Step 3: Solve the Integrals
From the left side, we have:
\[ -\sqrt{1 - y^2} = x + C \]
Rearranging gives:
\[ \sqrt{1 - y^2} = -x - C \]

### Step 4: Square Both Sides
Squaring both sides yields:
\[ 1 - y^2 = (x + C)^2 \]
Rearranging gives:
\[ y^2 = 1 - (x + C)^2 \]

### Step 5: Identify the Family of Circles
This equation can be rewritten as:
\[ (x + C)^2 + y^2 = 1 \]
This represents a family of circles with a fixed radius of 1 and variable centers at \((-C, 0)\).

### Conclusion
Thus, the differential equation \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{y} \) determines a family of circles with a fixed radius of 1 and variable centers along the x-axis.

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