Solution of the differential equation
`x((dy)/(dx))^(2)+2sqrt(xy)(dy)/(dx)+y=0` is

Options

A. `x+y=a`
B. `sqrtx-sqrty=a`
C. `x^(2)+y^(2)=a^(2)`
D. `sqrtx+sqrty=sqrta`

Explanation

To solve the differential equation

\[ x\left(\frac{dy}{dx}\right)^2 + 2\sqrt{xy}\frac{dy}{dx} + y = 0, \]

we can follow these steps:

### Step 1: Rewrite the equation
We can rewrite the equation in a more recognizable form. Notice that we can express the left-hand side as a perfect square:

\[ \left(\sqrt{x}\frac{dy}{dx} + \sqrt{y}\right)^2 = 0. \]

### Step 2: Set the perfect square to zero
Since the square of a term is zero, we can set the expression inside the square to zero:

\[ \sqrt{x}\frac{dy}{dx} + \sqrt{y} = 0. \]

### Step 3: Isolate \(\frac{dy}{dx}\)
Now, we can isolate \(\frac{dy}{dx}\):

\[ \sqrt{x}\frac{dy}{dx} = -\sqrt{y}. \]

Dividing both sides by \(\sqrt{x}\):

\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}}. \]

### Step 4: Separate variables
We can separate the variables \(y\) and \(x\):

\[ \frac{dy}{\sqrt{y}} = -\frac{dx}{\sqrt{x}}. \]

### Step 5: Integrate both sides
Now, we will integrate both sides. The left side can be integrated as follows:

\[ \int \frac{dy}{\sqrt{y}} = 2\sqrt{y}, \]

and the right side:

\[ -\int \frac{dx}{\sqrt{x}} = -2\sqrt{x}. \]

Thus, we have:

\[ 2\sqrt{y} = -2\sqrt{x} + C, \]

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

### Step 6: Simplify the equation
Dividing the entire equation by 2 gives:

\[ \sqrt{y} = -\sqrt{x} + \frac{C}{2}. \]

### Step 7: Rearranging the equation
Rearranging this, we can express it as:

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

Letting \(A = \frac{C}{2}\), we can write the final solution as:

\[ \sqrt{x} + \sqrt{y} = A, \]

where \(A\) is a constant.

### Conclusion
The solution of the differential equation is:

\[ \sqrt{x} + \sqrt{y} = A. \]