what is the solution of the differential equation
` (dx)/(dy) +x/y -y^(2) =0` ?
where c is an arbitraty constant .

Options

A. `xy=x^(2) +c`
B. `xy =y^(2)+c`
C. `4xy= y^(2) +c`
D. ` 3xy = y^(3)+c`

Explanation

To solve the differential equation

\[ \frac{dx}{dy} + \frac{x}{y} - y^2 = 0, \]

we will follow these steps:

### Step 1: Rewrite the Equation
We can rearrange the equation to isolate \(\frac{dx}{dy}\):

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

### Step 2: Identify \(p(y)\) and \(q(y)\)
This is a linear first-order differential equation of the form:

\[ \frac{dx}{dy} + p(y)x = q(y), \]

where \(p(y) = \frac{1}{y}\) and \(q(y) = y^2\).

### Step 3: Find the Integrating Factor
The integrating factor \(\mu(y)\) is given by:

\[ \mu(y) = e^{\int p(y) \, dy} = e^{\int \frac{1}{y} \, dy} = e^{\log |y|} = |y|. \]

Since \(y\) is positive in the context of this problem, we can simply use \(\mu(y) = y\).

### Step 4: Multiply the Equation by the Integrating Factor
Now we multiply the entire differential equation by the integrating factor \(y\):

\[ y \frac{dx}{dy} + x = y^3. \]

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

\[ \frac{d}{dy}(xy) = y^3. \]

### Step 6: Integrate Both Sides
Now we integrate both sides with respect to \(y\):

\[ \int \frac{d}{dy}(xy) \, dy = \int y^3 \, dy. \]

This gives us:

\[ xy = \frac{y^4}{4} + C, \]

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

### Step 7: Solve for \(x\)
Finally, we solve for \(x\):

\[ x = \frac{y^4}{4y} + \frac{C}{y} = \frac{y^3}{4} + \frac{C}{y}. \]

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

\[ x = \frac{y^3}{4} + \frac{C}{y}, \]

where \(C\) is an arbitrary constant.

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