`x(dy)/(dx)+(y^(2))/(x)=y`

Options

A. `(y)/(x)+logx+c`
B. `(y)/(x)=logy+c`
C. `-(x)/(y)=logx+c`
D. `y=xlogy+c`

Answer

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

### Step 1: Rearrange the Equation
First, we can rearrange the equation to isolate the term involving \(\frac{dy}{dx}\):
\[ x \frac{dy}{dx} = y - \frac{y^2}{x} \]

### Step 2: Divide by \(x\)
Next, we divide the entire equation by \(x\):
\[ \frac{dy}{dx} = \frac{y}{x} - \frac{y^2}{x^2} \]

### Step 3: Rewrite the Equation
Now, we can rewrite the equation as:
\[ \frac{dy}{dx} + \frac{y^2}{x^2} = \frac{y}{x} \]

### Step 4: Identify \(p(x)\) and \(q(x)\)
We can identify \(p(x)\) and \(q(x)\) from the standard form of the linear differential equation:
- \(p(x) = \frac{1}{x}\)
- \(q(x) = \frac{1}{x^2}\)

### Step 5: Find the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \]
Since \(x\) is positive in this context, we have:
\[ I(x) = x \]

### Step 6: Multiply the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor \(x\):
\[ x \frac{dy}{dx} + \frac{y^2}{x} = y \]

### Step 7: Integrate
This can be rewritten as:
\[ \frac{d}{dx}(xy) = \int \frac{1}{x} \, dx \]
Integrating both sides gives:
\[ xy = \ln |x| + C \]

### Step 8: Solve for \(y\)
Finally, we can solve for \(y\):
\[ y = \frac{\ln |x| + C}{x} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = \frac{\ln |x| + C}{x} \]