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

Options

A. `sin^(-1)x-sin^(-1)y=c`
B. `sin^(-1)y-sin^(-1)x=c`
C. `sin^(-1)(y-x)=c`
D. `x-y=c(1+xy)`

Answer

To solve the differential equation

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

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate \(\frac{dy}{dx}\):

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

**Hint:** When rearranging differential equations, aim to isolate the derivative on one side.

### Step 2: Cross Multiplication
Next, we can cross-multiply to separate the variables \(y\) and \(x\):

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

**Hint:** Cross multiplication is a useful technique to separate variables in a differential equation.

### Step 3: Integrating Both Sides
Now, we will integrate both sides. The integral of \(\frac{1}{\sqrt{1 - \theta^2}} d\theta\) is \(\sin^{-1}(\theta)\):

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

This gives us:

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

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

**Hint:** Remember the integral of \(\frac{1}{\sqrt{1 - \theta^2}}\) leads to the inverse sine function.

### Step 4: Rearranging the Result
We can rearrange the equation to express the relationship between \(y\) and \(x\):

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

**Hint:** Rearranging the equation can help in identifying the relationship between the variables.

### Final Result
The general solution of the differential equation is:

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