The general solution of the differential equation `ydx+(x+2y^(2))dy=0` is

Options

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

Explanation

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

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ y \, dx + (x + 2y^2) \, dy = 0 \]
We can rearrange this as:
\[ y \, dx = - (x + 2y^2) \, dy \]

### Step 2: Separating Variables
Now, we can separate the variables:
\[ \frac{dx}{x + 2y^2} = -\frac{dy}{y} \]

### Step 3: Integrating Both Sides
Next, we integrate both sides. The left side will be integrated with respect to \( x \) and the right side with respect to \( y \):
\[ \int \frac{dx}{x + 2y^2} = -\int \frac{dy}{y} \]

### Step 4: Performing the Integrations
The left side integrates to:
\[ \ln |x + 2y^2| + C_1 \]
And the right side integrates to:
\[ -\ln |y| + C_2 \]
So, we have:
\[ \ln |x + 2y^2| = -\ln |y| + C \]
where \( C = C_2 - C_1 \).

### Step 5: Simplifying the Equation
We can exponentiate both sides to eliminate the logarithm:
\[ |x + 2y^2| = \frac{C}{|y|} \]

### Step 6: Rearranging to Find the General Solution
Multiplying both sides by \( |y| \) gives:
\[ |y| (x + 2y^2) = C \]
This can be simplified to:
\[ 3xy + 2y^3 = C \]

### Conclusion
Thus, the general solution of the differential equation is:
\[ 3xy + 2y^3 = C \]

---