The solution of the differential equation `(1-x^(2))(dy)/(dx)-xy=1` is (where, `|x|lt1, x in R` and C is an arbitrary constant)

Options

A. `y(1-x^(2))=tan^(-1)x+C`
B. `ysqrt(1-x^(2))=tan^(-1)x+C`
C. `ysqrt(1-x^(2))=sin^(-1)(x)+C`
D. `y.(1-x^(2))=sin^(-1)x+C`

Answer

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

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ (1 - x^2) \frac{dy}{dx} - xy = 1 \]
We can rearrange it to isolate \(\frac{dy}{dx}\):
\[ (1 - x^2) \frac{dy}{dx} = xy + 1 \]
Now, divide both sides by \(1 - x^2\):
\[ \frac{dy}{dx} = \frac{xy + 1}{1 - x^2} \]

### Step 2: Expressing in Standard Form
Next, we can express this in the standard form of a linear differential equation:
\[ \frac{dy}{dx} + \frac{xy}{1 - x^2} = \frac{1}{1 - x^2} \]
Here, we identify:
- \(p(x) = \frac{x}{1 - x^2}\)
- \(q(x) = \frac{1}{1 - x^2}\)

### Step 3: Finding the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int p(x) \, dx} \]
Calculating the integral:
\[ \int \frac{x}{1 - x^2} \, dx \]
We can use the substitution \(t = 1 - x^2\), which gives \(dt = -2x \, dx\) or \(dx = -\frac{dt}{2x}\). Thus, we rewrite the integral:
\[ \int \frac{x}{t} \left(-\frac{dt}{2x}\right) = -\frac{1}{2} \int \frac{1}{t} \, dt = -\frac{1}{2} \ln |t| + C = -\frac{1}{2} \ln |1 - x^2| + C \]
Thus, the integrating factor is:
\[ I(x) = e^{-\frac{1}{2} \ln |1 - x^2|} = |1 - x^2|^{-\frac{1}{2}} = \frac{1}{\sqrt{1 - x^2}} \]

### Step 4: Multiplying through by the Integrating Factor
Now, multiply the entire differential equation by the integrating factor:
\[ \frac{1}{\sqrt{1 - x^2}} \frac{dy}{dx} + \frac{xy}{\sqrt{1 - x^2}} = \frac{1}{\sqrt{1 - x^2}(1 - x^2)} \]

### Step 5: Integrating Both Sides
The left-hand side can be rewritten as:
\[ \frac{d}{dx}\left(y \sqrt{1 - x^2}\right) \]
Thus, we have:
\[ \frac{d}{dx}\left(y \sqrt{1 - x^2}\right) = \frac{1}{\sqrt{1 - x^2}(1 - x^2)} \]
Integrating both sides gives:
\[ y \sqrt{1 - x^2} = \int \frac{1}{\sqrt{1 - x^2}(1 - x^2)} \, dx + C \]

### Step 6: Solving the Integral
The integral \(\int \frac{1}{\sqrt{1 - x^2}(1 - x^2)} \, dx\) can be recognized as:
\[ \sin^{-1}(x) + C \]
Thus, we have:
\[ y \sqrt{1 - x^2} = \sin^{-1}(x) + C \]

### Step 7: Final Solution
Finally, we can express \(y\):
\[ y = \frac{\sin^{-1}(x) + C}{\sqrt{1 - x^2}} \]

### Conclusion
The solution of the differential equation is:
\[ y = \frac{\sin^{-1}(x) + C}{\sqrt{1 - x^2}} \]
where \(|x| < 1\) and \(C\) is an arbitrary constant.