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 \( ydx + (x + 2y^2)dy = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ ydx + (x + 2y^2)dy = 0 \]
We can rewrite this as:
\[ ydx + xdy + 2y^2dy = 0 \]

### Step 2: Identify the total differential
Notice that the first two terms \( ydx + xdy \) can be recognized as the total differential of the product \( xy \). Thus, we can express it as:
\[ d(xy) + 2y^2dy = 0 \]

### Step 3: Separate variables
Now, we can rearrange the equation:
\[ d(xy) = -2y^2dy \]

### Step 4: Integrate both sides
Integrating both sides:
\[ \int d(xy) = \int -2y^2 dy \]
The left side integrates to \( xy \), and the right side integrates to:
\[ -2 \cdot \frac{y^3}{3} = -\frac{2}{3}y^3 \]
Thus, we have:
\[ xy = -\frac{2}{3}y^3 + C \]
where \( C \) is the constant of integration.

### Step 5: Rearranging the equation
To express the solution in a standard form, we can multiply through by 3:
\[ 3xy + 2y^3 = 3C \]
Letting \( C' = 3C \) (which is still a constant), we can write:
\[ 3xy + 2y^3 + C' = 0 \]

### Final General Solution
Thus, the general solution of the given differential equation is:
\[ 3xy + 2y^3 + C = 0 \]