`(dy)/(dx) = sqrt((1-y^(2))/(1-x^(2)))`

Answer

To solve the differential equation \(\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}\), we can separate the variables and integrate both sides. Here’s a step-by-step solution:

### Step 1: Separate the Variables
We start with the given equation:

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

We can rewrite this as:

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

### Step 2: Integrate Both Sides
Now we integrate both sides. The left side is integrated with respect to \(y\) and the right side with respect to \(x\):

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

### Step 3: Solve the Integrals
The integral \(\int \frac{dy}{\sqrt{1 - y^2}}\) is known to be:

\[ \arcsin(y) + C_1 \]

Similarly, the integral \(\int \frac{dx}{\sqrt{1 - x^2}}\) is:

\[ \arcsin(x) + C_2 \]

Thus, we can write:

\[ \arcsin(y) = \arcsin(x) + C \]

where \(C = C_2 - C_1\).

### Step 4: Solve for \(y\)
To express \(y\) in terms of \(x\), we take the sine of both sides:

\[ y = \sin(\arcsin(x) + C) \]

### Step 5: Use the Sine Addition Formula
Using the sine addition formula, we can express \(y\) as:

\[ y = \sin(\arcsin(x)) \cos(C) + \cos(\arcsin(x)) \sin(C) \]

Since \(\sin(\arcsin(x)) = x\) and \(\cos(\arcsin(x)) = \sqrt{1 - x^2}\), we have:

\[ y = x \cos(C) + \sqrt{1 - x^2} \sin(C) \]

### Final Solution
Letting \(k = \cos(C)\) and \(m = \sin(C)\), we can express the final solution as:

\[ y = kx + m\sqrt{1 - x^2} \]

where \(k\) and \(m\) are constants determined by initial conditions.

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