The Solution of the equation
`(1+x^(2)) (dy)/(dx)+2xy -4x^(2)=0`

Options

A. `y(1+x^(2))=x^(3)+c`
B. `y(1+x^(2))=2x+c`
C. `y(1+x^(2))=(4)/(3)x^(3)+c`
D. None

Answer

To solve the differential equation

\[ (1+x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate \(\frac{dy}{dx}\):

\[ (1+x^2) \frac{dy}{dx} = -2xy + 4x^2. \]

Dividing through by \(1+x^2\):

\[ \frac{dy}{dx} = \frac{-2xy + 4x^2}{1+x^2}. \]

### Step 2: Expressing in Standard Linear Form
We can express this in the standard linear form \(\frac{dy}{dx} + P(x)y = Q(x)\):

\[ \frac{dy}{dx} + \frac{2x}{1+x^2} y = \frac{4x}{1+x^2}. \]

Here, we identify \(P(x) = \frac{2x}{1+x^2}\) and \(Q(x) = \frac{4x}{1+x^2}\).

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

\[ I(x) = e^{\int P(x) \, dx} = e^{\int \frac{2x}{1+x^2} \, dx}. \]

To compute the integral:

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

Thus, the integrating factor becomes:

\[ I(x) = e^{\ln(1+x^2)} = 1+x^2. \]

### Step 4: Multiplying through by the Integrating Factor
Now we multiply the entire differential equation by the integrating factor:

\[ (1+x^2) \frac{dy}{dx} + 2xy = 4x. \]

### Step 5: Recognizing the Left-Hand Side as a Derivative
The left-hand side can be recognized as the derivative of a product:

\[ \frac{d}{dx}(y(1+x^2)) = 4x. \]

### Step 6: Integrating Both Sides
Integrate both sides with respect to \(x\):

\[ y(1+x^2) = \int 4x \, dx = 2x^2 + C, \]

where \(C\) is the constant of integration.

### Step 7: Solving for \(y\)
Now, we solve for \(y\):

\[ y = \frac{2x^2 + C}{1+x^2}. \]

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

\[ y = \frac{2x^2 + C}{1+x^2}. \]