The solution of the differential equation `2ydx+xdy=2x sqrtydx` is (where, C is an arbitrary constant)

Options

A. `xsqrty=x+C`
B. `xsqrty=(x^(2))/(2)+C`
C. `(x)/(sqrty)=x+C`
D. `xy=C`

Answer

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

### Step 1: Rearranging the Equation
We start with the given equation:
\[ 2y \, dx + x \, dy = 2x \sqrt{y} \, dx \]
We can rearrange this to isolate \( dy \) and \( dx \):
\[ x \, dy = 2x \sqrt{y} \, dx - 2y \, dx \]
This simplifies to:
\[ x \, dy = (2x \sqrt{y} - 2y) \, dx \]

### Step 2: Dividing by \( 2\sqrt{y} \)
Next, we divide the entire equation by \( 2\sqrt{y} \):
\[ \frac{2y}{2\sqrt{y}} \, dx + \frac{x}{2\sqrt{y}} \, dy = \frac{2x\sqrt{y}}{2\sqrt{y}} \, dx \]
This simplifies to:
\[ \sqrt{y} \, dx + \frac{x}{2\sqrt{y}} \, dy = x \, dx \]

### Step 3: Identifying the Total Differential
Now, we can recognize that the left side resembles the total differential of the product \( x\sqrt{y} \):
\[ d(x\sqrt{y}) = \sqrt{y} \, dx + x \cdot \frac{1}{2\sqrt{y}} \, dy \]
Thus, we can rewrite our equation as:
\[ d(x\sqrt{y}) = x \, dx \]

### Step 4: Integrating Both Sides
Now we integrate both sides:
\[ \int d(x\sqrt{y}) = \int x \, dx \]
The left side integrates to:
\[ x\sqrt{y} = \frac{x^2}{2} + C \]
where \( C \) is the constant of integration.

### Step 5: Final Form
Rearranging gives us the final solution:
\[ x\sqrt{y} = \frac{x^2}{2} + C \]

### Summary of the Solution
The solution of the differential equation is:
\[ x\sqrt{y} = \frac{x^2}{2} + C \]

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