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

Options

A. `2sin^(-1)y=x sqrt(1-x^(2))+sin^(-1)x+c`
B. `cos^(-1)y=x.cos^(-1)x+c`
C. `sin^(-1)y=(1)/(2).sin^(-1)x+c`
D. `2.sin^(-1)y=x.sqrt(1-y^(2))+c`

Answer

To find the general solution of the differential equation

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

we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:

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

We can factor the expression under the square root:

\[ \sqrt{(1 - x^2)(1 - y^2)}. \]

Thus, we can rewrite the equation as:

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

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

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

### Step 3: Integrate both sides
Now we integrate both sides:

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

The left-hand side integrates to:

\[ \sin^{-1}(y) + C_1, \]

and the right-hand side can be solved using the formula for the integral of \(\sqrt{a^2 - x^2}\):

\[ \int \sqrt{1 - x^2} \, dx = \frac{x}{2} \sqrt{1 - x^2} + \frac{1}{2} \sin^{-1}(x) + C_2. \]

### Step 4: Combine results
Combining the results from both integrals, we have:

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

where \(C\) is a constant that combines \(C_1\) and \(C_2\).

### Step 5: Simplify the equation
To express the final result clearly, we can multiply through by 2:

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

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

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

where \(C\) is an arbitrary constant.

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