The solution of differential equation
` (x^(2)-1) (dy)/(dx) + 2xy = 1/(x^(2)-1)` is

Options

A. ` y(x^(2)-1)=1/2 log |(x-1)/(x+1)|+C`
B. ` y (x^(2)+1)=1/2 log |(x-1)/(x+1)|+C`
C. ` y (x^(2)+1)=1/3 log |(x-1)/(x+1)|+C`
D. `y ( x^(2)-1)=1/3 log |(x-1)/(x+1)|+C`

Explanation

To solve the differential equation

\[ (x^2 - 1) \frac{dy}{dx} + 2xy = \frac{1}{x^2 - 1} \]

we will follow these steps:

### Step 1: Rewrite the equation
First, we will divide the entire equation by \(x^2 - 1\) to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} + \frac{2xy}{x^2 - 1} = \frac{1}{(x^2 - 1)^2} \]

### Step 2: Identify \(p(x)\) and \(q(x)\)
In this linear differential equation, we identify:
- \(p(x) = \frac{2x}{x^2 - 1}\)
- \(q(x) = \frac{1}{(x^2 - 1)^2}\)

### Step 3: Find the integrating factor
The integrating factor \(I(x)\) is given by:

\[ I(x) = e^{\int p(x) \, dx} = e^{\int \frac{2x}{x^2 - 1} \, dx} \]

To solve the integral, we can use substitution. Let \(t = x^2 - 1\), then \(dt = 2x \, dx\) or \(dx = \frac{dt}{2x}\). The integral becomes:

\[ \int \frac{2x}{t} \cdot \frac{dt}{2x} = \int \frac{1}{t} \, dt = \ln |t| + C = \ln |x^2 - 1| + C \]

Thus, the integrating factor is:

\[ I(x) = e^{\ln |x^2 - 1|} = |x^2 - 1| \]

### Step 4: Multiply the equation by the integrating factor
Now, we multiply the entire differential equation by the integrating factor:

\[ |x^2 - 1| \frac{dy}{dx} + |x^2 - 1| \frac{2xy}{x^2 - 1} = |x^2 - 1| \frac{1}{(x^2 - 1)^2} \]

This simplifies to:

\[ |x^2 - 1| \frac{dy}{dx} + 2xy = \frac{1}{x^2 - 1} \]

### Step 5: Rewrite the left-hand side
The left-hand side can be expressed as the derivative of a product:

\[ \frac{d}{dx}(y |x^2 - 1|) = \frac{1}{(x^2 - 1)^2} \]

### Step 6: Integrate both sides
Now we integrate both sides:

\[ \int \frac{d}{dx}(y |x^2 - 1|) \, dx = \int \frac{1}{(x^2 - 1)^2} \, dx \]

The left-hand side simplifies to:

\[ y |x^2 - 1| = \int \frac{1}{(x^2 - 1)^2} \, dx + C \]

### Step 7: Solve the integral on the right-hand side
The integral \(\int \frac{1}{(x^2 - 1)^2} \, dx\) can be solved using partial fractions or trigonometric substitution. The result is:

\[ \int \frac{1}{(x^2 - 1)^2} \, dx = -\frac{1}{2(x^2 - 1)} + C \]

### Step 8: Solve for \(y\)
Now, substituting back, we have:

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

Finally, we can express \(y\):

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

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

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