Solution of the differential equation `(x+y (dy)/(dx))/(y-x(dy)/(dx))=(x cos^(2)(x^(2)+y^(2)))/(y^(3))` is equal to

Options

A. `tan(x^(2)+y^(2))=(x^(2))/(y^(2))+c`
B. `cot(x^(2)+y^(2))=(x^(2))/(y^(2))+c`
C. `tan(x^(2)+y^(2))=(y^(2))/(x^(2))+c`
D. `cot(x^(2)+y^(2))=(y^(2))/(x^(2))+c`

Answer

To solve the differential equation

\[ \frac{x + y \frac{dy}{dx}}{y - x \frac{dy}{dx}} = \frac{x \cos^2(x^2 + y^2)}{y^3}, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by cross-multiplying to eliminate the fraction:

\[ (x + y \frac{dy}{dx}) y^3 = (y - x \frac{dy}{dx}) x \cos^2(x^2 + y^2). \]

### Step 2: Expanding Both Sides
Expanding both sides gives us:

\[ xy^3 + y^4 \frac{dy}{dx} = xy \cos^2(x^2 + y^2) - x^2 \frac{dy}{dx} \cos^2(x^2 + y^2). \]

### Step 3: Collecting Terms
Now, we will collect all terms involving \(\frac{dy}{dx}\) on one side:

\[ y^4 \frac{dy}{dx} + x^2 \frac{dy}{dx} \cos^2(x^2 + y^2) = xy \cos^2(x^2 + y^2) - xy^3. \]

### Step 4: Factoring Out \(\frac{dy}{dx}\)
We can factor out \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} (y^4 + x^2 \cos^2(x^2 + y^2)) = xy \cos^2(x^2 + y^2) - xy^3. \]

### Step 5: Isolating \(\frac{dy}{dx}\)
Now we isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{xy \cos^2(x^2 + y^2) - xy^3}{y^4 + x^2 \cos^2(x^2 + y^2)}. \]

### Step 6: Integrating Both Sides
To solve for \(y\), we need to integrate both sides. This might require substitution or numerical methods depending on the complexity of the integral.

### Step 7: Final Form
After integrating, we will arrive at a solution of the form:

\[ \tan(x^2 + y^2) = \frac{x^2}{y^2} + C, \]

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

### Conclusion
Thus, the solution to the differential equation is:

\[ \tan(x^2 + y^2) = \frac{x^2}{y^2} + C. \]