The general solution of the differential equation `[2 sqrt( xy ) -x] dy + y dx= 0` is (Here x,`y gt 0 ` )

Options

A. `log + sqrt((y)/(x)=c`
B. `log - sqrt((x)/(y))=C`
C. `log y + sqrt((x ) /(y) )=C`
D. none of these

Answer

To solve the differential equation \( [2 \sqrt{xy} - x] dy + y dx = 0 \), we will follow these steps:

### Step 1: Rewrite the Equation
Start by rewriting the given equation in a more manageable form:
\[ 2 \sqrt{xy} \, dy - x \, dy + y \, dx = 0 \]
This can be rearranged as:
\[ y \, dx - x \, dy = -2 \sqrt{xy} \, dy \]

### Step 2: Use the Total Differential
We can use the total differential formula:
\[ \frac{dy}{y} = \frac{dx}{x} - \frac{2 \sqrt{xy}}{y^2} \, dy \]
This implies:
\[ \frac{dy}{y} = \frac{dx}{x} - \frac{2 \sqrt{x}}{y} \, dy \]

### Step 3: Isolate Terms
Rearranging gives:
\[ \frac{dy}{y} + \frac{2 \sqrt{x}}{y} \, dy = \frac{dx}{x} \]
Now, factor out \( dy \):
\[ \left( \frac{1}{y} + \frac{2 \sqrt{x}}{y} \right) dy = \frac{dx}{x} \]

### Step 4: Integrate Both Sides
Integrating both sides:
\[ \int \left( \frac{1 + 2 \sqrt{x}}{y} \right) dy = \int \frac{dx}{x} \]
This leads to:
\[ \log y + 2 \sqrt{x} = \log x + C \]
Where \( C \) is the integration constant.

### Step 5: Simplify the Result
We can rearrange the equation:
\[ \log y - \log x + 2 \sqrt{x} = C \]
This can be expressed as:
\[ \log \left( \frac{y}{x} \right) + 2 \sqrt{x} = C \]

### Step 6: Exponentiate to Solve for y
Exponentiating both sides gives:
\[ \frac{y}{x} e^{2 \sqrt{x}} = e^C \]
Let \( K = e^C \), then:
\[ y = Kx e^{-2 \sqrt{x}} \]

### Final General Solution
Thus, the general solution of the differential equation is:
\[ y = C x e^{-2 \sqrt{x}} \]
where \( C \) is a constant.