Solve the differential equation : `(dy)/(dx) =1 + x^(2)+y^(2)+x^(2)y^(2)` .

Answer

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

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ \frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2 \]
We can rearrange this equation to group terms involving \(y\):
\[ \frac{dy}{dx} = (1 + y^2) + x^2(1 + y^2) \]
This can be factored as:
\[ \frac{dy}{dx} = (1 + y^2)(1 + x^2) \]

### Step 2: Separating Variables
Now we can separate the variables \(y\) and \(x\):
\[ \frac{dy}{1 + y^2} = (1 + x^2)dx \]

### Step 3: Integrating Both Sides
Next, we integrate both sides:
\[ \int \frac{dy}{1 + y^2} = \int (1 + x^2)dx \]
The left side integrates to:
\[ \int \frac{dy}{1 + y^2} = \tan^{-1}(y) + C_1 \]
The right side integrates to:
\[ \int (1 + x^2)dx = x + \frac{x^3}{3} + C_2 \]

### Step 4: Combining Constants
We can combine the constants \(C_1\) and \(C_2\) into a single constant \(C\):
\[ \tan^{-1}(y) = x + \frac{x^3}{3} + C \]

### Step 5: Solving for \(y\)
To express \(y\) in terms of \(x\), we take the tangent of both sides:
\[ y = \tan\left(x + \frac{x^3}{3} + C\right) \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = \tan\left(x + \frac{x^3}{3} + C\right) \]

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