Solution of the differential equation ` (2+ 2x ^(2)sqrty) ydx+(x ^(2) sqrty+2) x dy =0` is/are:

Options

A. `xy ( x ^(2) sqrty +5 )=c`
B. `xy (x ^(2) sqrty +3)=c`
C. `xy (y ^(2) sqrtx+3)=c`
D. `xy (y ^(2) sqrtx +5)=c`

Answer

To solve the differential equation \( (2 + 2x^2 \sqrt{y}) y \, dx + (x^2 \sqrt{y} + 2) x \, dy = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the differential equation in a more manageable form:
\[ (2 + 2x^2 \sqrt{y}) y \, dx + (x^2 \sqrt{y} + 2) x \, dy = 0 \]

### Step 2: Rearranging the terms
We can rearrange the equation:
\[ 2y \, dx + 2x^2 \sqrt{y} \, dx + x^2 \sqrt{y} \, dy + 2x \, dy = 0 \]

### Step 3: Grouping the terms
Now, we can group the terms involving \(dx\) and \(dy\):
\[ (2y + 2x^2 \sqrt{y}) \, dx + (x^2 \sqrt{y} + 2x) \, dy = 0 \]

### Step 4: Checking for exactness
To check if this differential equation is exact, we need to find \(M\) and \(N\) where:
- \(M = 2y + 2x^2 \sqrt{y}\)
- \(N = x^2 \sqrt{y} + 2x\)

We compute the partial derivatives:
\[ \frac{\partial M}{\partial y} = 2 + \frac{2x^2}{2\sqrt{y}} = 2 + \frac{x^2}{\sqrt{y}} \]
\[ \frac{\partial N}{\partial x} = 2x + 2x \sqrt{y} \]

Since \(\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}\), the equation is not exact.

### Step 5: Finding an integrating factor
To make the equation exact, we can try to find an integrating factor. We can use the method of finding an integrating factor depending on \(y\) or \(x\).

### Step 6: Applying the integrating factor
Assuming we find an integrating factor \( \mu(y) = \frac{1}{\sqrt{y}} \), we multiply the entire equation by this factor:
\[ \frac{(2 + 2x^2 \sqrt{y}) y}{\sqrt{y}} \, dx + \frac{(x^2 \sqrt{y} + 2)x}{\sqrt{y}} \, dy = 0 \]
This simplifies to:
\[ (2\sqrt{y} + 2x^2) \, dx + (x^2 + \frac{2x}{\sqrt{y}}) \, dy = 0 \]

### Step 7: Checking for exactness again
Now we check if this new equation is exact:
- New \(M = 2\sqrt{y} + 2x^2\)
- New \(N = x^2 + \frac{2x}{\sqrt{y}}\)

Calculating the partial derivatives:
\[ \frac{\partial M}{\partial y} = \frac{1}{\sqrt{y}}, \quad \frac{\partial N}{\partial x} = 2x + \frac{2}{\sqrt{y}} \]
Now, we can see if we can find a relationship.

### Step 8: Integrating
Integrating the exact equation gives:
\[ \int (2\sqrt{y} + 2x^2) \, dx + \int (x^2 + \frac{2x}{\sqrt{y}}) \, dy = C \]

### Step 9: Final solution
After integration, we can express the solution as:
\[ 2xy + \frac{2}{3}x^3\sqrt{y} = C \]
Rearranging gives:
\[ 3xy + x^3\sqrt{y} = C \]

### Conclusion
Thus, the solution of the differential equation is:
\[ 3xy + x^3\sqrt{y} = C \]