The solution of `xdy+ydx+2x^(3)dx=0` is

Options

A. `xy+x^(4)=c`
B. `xy+(1)/(2)x^(4)=c`
C. `(x^(2))/(y)+(x^(4))/(4)=c`
D. None of these

Answer

To solve the differential equation \( x \, dy + y \, dx + 2x^3 \, dx = 0 \), we will 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 it to isolate the \( dy \) and \( dx \) terms:
\[ x \, dy + y \, dx = -2x^3 \, dx \]

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

### Step 3: Integrating Both Sides
Now we will 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 the Solution
To express the solution in a standard form, we can rearrange it:
\[ xy + \frac{1}{2} x^4 = C \]

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