The general solution of the differential equation `x dy+ydx+2x^(3)dx = 0`, is

Options

A. `xy+x^(4)=C`
B. `xy+(1)/(2)x^(4)=C`
C. `x+y+(1)/(2)x^(4)=0`
D. `xy-(1)/(2)x^(4)C`

Explanation

To solve the differential equation \( x \, dy + y \, dx + 2x^3 \, dx = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x \, dy + y \, dx + 2x^3 \, dx = 0 \]
We can rearrange this to isolate the \( dy \) term:
\[ x \, dy + y \, dx = -2x^3 \, dx \]

### Step 2: Identifying the Total Differential
Notice that the left-hand side can be recognized as the total differential of the product \( xy \):
\[ d(xy) = x \, dy + y \, dx \]
Thus, we can rewrite the equation as:
\[ d(xy) = -2x^3 \, dx \]

### Step 3: Integrating Both Sides
Now we integrate both sides. The left side integrates to:
\[ \int d(xy) = xy \]
For the right side, we factor out the constant:
\[ \int -2x^3 \, dx = -2 \cdot \frac{x^4}{4} = -\frac{1}{2} x^4 \]
So, we have:
\[ xy = -\frac{1}{2} x^4 + C \]
where \( C \) is the constant of integration.

### Step 4: Rearranging for the General Solution
Rearranging gives us the general solution:
\[ xy + \frac{1}{2} x^4 = C \]

### Final Solution
Thus, the general solution of the differential equation is:
\[ xy + \frac{1}{2} x^4 = C \]

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