Solutionof the differential equation `ydx-xdy+xsqrt(xy)dy=0` is

Options

A. `(x^(2)+y^(2))/2 + tan^(-1)sqrt(y/x)=c`
B. `(x^(2)+y^(2))/2 + 2tan^(-1)sqrt(x/y)=C`
C. `(x^(2)+y^(2))/2+2cot^(-1)sqrt(x/y)=c`
D. None of these

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ y \, dx - x \, dy + x \sqrt{xy} \, dy = 0 \]
We can rearrange it as:
\[ y \, dx + (x \sqrt{xy} - x) \, dy = 0 \]

### Step 2: Factor out common terms
We can factor out \( x \) from the dy term:
\[ y \, dx + x (\sqrt{xy} - 1) \, dy = 0 \]

### Step 3: Isolate dy
Rearranging gives:
\[ y \, dx = -x (\sqrt{xy} - 1) \, dy \]
Now, divide both sides by \( x \):
\[ \frac{y}{x} \, dx = -(\sqrt{xy} - 1) \, dy \]

### Step 4: Substitute and simplify
Let \( a = \frac{y}{x} \), then \( y = ax \) and \( dy = a \, dx + x \, da \). Substituting this into the equation gives:
\[ \frac{y}{x} \, dx = -(\sqrt{ax^2} - 1) \, (a \, dx + x \, da) \]
This simplifies to:
\[ a \, dx = -(\sqrt{a} x - 1) (a \, dx + x \, da) \]

### Step 5: Separate variables
We can separate variables to integrate:
\[ \frac{a \, dx}{-a(\sqrt{a} x - 1)} = da \]

### Step 6: Integrate both sides
Now we integrate both sides:
\[ \int \frac{1}{-\sqrt{a} x + 1} \, dx = \int da \]
This will yield:
\[ -\ln |-\sqrt{a} x + 1| = a + C \]

### Step 7: Solve for y
Substituting back \( a = \frac{y}{x} \):
\[ -\ln |-\sqrt{\frac{y}{x}} x + 1| = \frac{y}{x} + C \]
This can be rearranged to express \( y \) in terms of \( x \).

### Final Solution
After simplifying, we arrive at the implicit solution:
\[ \sqrt{xy} - C = 2\sqrt{y} \]
This can be rearranged to find \( y \) explicitly if needed.