The solution of `(dy)/(dx) + sqrt(((1-y^(2))/(1-x^(2))))=0`is

Options

A. `tan^(-1)x +cot^(-1)x = C `
B. `sin^(-1)x + sin^(-1)y=C`
C. `sec^(-1)x+"cosec"^(-1)x=C`
D. None of these

Explanation

To solve the differential equation

\[ \frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0, \]

we can follow these steps:

### Step 1: Rearrange the equation
First, we can isolate \(\frac{dy}{dx}\):

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

### Step 2: Separate variables
Next, we can separate the variables \(y\) and \(x\):

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

### Step 3: Integrate both sides
Now we will integrate both sides. The left side integrates to \(\sin^{-1}(y)\) and the right side integrates to \(-\sin^{-1}(x)\):

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

This gives us:

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

where \(C\) is the constant of integration.

### Step 4: Rearranging the equation
Now we can rearrange the equation to express it in a more standard form:

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

### Final Solution
Thus, the solution of the differential equation is:

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