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 by rearranging the given equation into a more manageable form. We can express it as:
\[ x \, dy + (y + 2x^3) \, dx = 0 \]
This can be rewritten as:
\[ x \frac{dy}{dx} + y + 2x^3 = 0 \]

### Step 2: Dividing by \( x \)
Next, we divide the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ \frac{dy}{dx} + \frac{y}{x} + 2x^2 = 0 \]
This is now in the standard form of a first-order linear differential equation:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = \frac{1}{x} \) and \( Q(x) = -2x^2 \).

### Step 3: Finding the Integrating Factor
To solve this equation, we need to find the integrating factor \( \mu(x) \):
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\log |x|} = |x| \]
Since we are assuming \( x > 0 \), we can simplify this to:
\[ \mu(x) = x \]

### Step 4: Multiplying the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y + 2x^3 = 0 \]
This simplifies to:
\[ \frac{d}{dx}(xy) = -2x^3 \]

### Step 5: Integrating Both Sides
Next, we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(xy) \, dx = \int -2x^3 \, dx \]
This gives us:
\[ xy = -\frac{2}{4}x^4 + C \]
or
\[ xy = -\frac{1}{2}x^4 + C \]

### Step 6: Final Solution
Rearranging the equation gives us the final solution:
\[ y = -\frac{1}{2}x^3 + \frac{C}{x} \]

### Summary of the Solution
The solution to the differential equation \( x \, dy + y \, dx + 2x^3 \, dx = 0 \) is:
\[ y = -\frac{1}{2}x^3 + \frac{C}{x} \]