If `(dy)/(dx) =(1+y^(2))/((1+x^(2))xy)`, then its solution is ……..

Answer

To solve the differential equation

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

we will use the method of separation of variables. Here’s the step-by-step solution:

### Step 1: Separate the Variables
We can rearrange the equation to separate the variables \(y\) and \(x\):

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

### Step 2: Rewrite the Left Side
The left side can be rewritten as:

\[ \left( \frac{1}{y} + y \right) dy. \]

Thus, we have:

\[ \left( \frac{1}{y} + y \right) dy = \frac{1}{(1 + x^2)x} \, dx. \]

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

\[ \int \left( \frac{1}{y} + y \right) dy = \int \frac{1}{(1 + x^2)x} \, dx. \]

The left side integrates to:

\[ \ln |y| + \frac{y^2}{2} + C_1, \]

and for the right side, we will use partial fractions to integrate:

\[ \frac{1}{(1 + x^2)x} = \frac{A}{x} + \frac{Bx + C}{1 + x^2}. \]

After finding the constants \(A\), \(B\), and \(C\) through algebraic manipulation, we can integrate the right side.

### Step 4: Solve for the Constants
After performing the partial fraction decomposition, we can integrate:

\[ \int \left( \frac{A}{x} + \frac{B}{1 + x^2} \right) dx = A \ln |x| + B \tan^{-1}(x) + C_2. \]

### Step 5: Combine and Simplify
Now we have:

\[ \ln |y| + \frac{y^2}{2} = A \ln |x| + B \tan^{-1}(x) + C, \]

where \(C = C_2 - C_1\) is a new constant.

### Step 6: Exponentiate to Eliminate Logarithm
To eliminate the logarithm, we exponentiate both sides:

\[ |y| = e^{A \ln |x| + B \tan^{-1}(x) + C - \frac{y^2}{2}}. \]

### Step 7: Final Form
This can be rearranged to give the final implicit solution of the differential equation.

The final solution can be expressed as:

\[ (1 + y^2)(1 + x^2) = Cx^2, \]

where \(C\) is a constant derived from the integration process.