The solution of the differential equation `xdy-ydx+3x^(2)y^(2)e^(x^(3))dx=0` is (where, c is an arbitrary constant)

Options

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

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ x dy - y dx = -3x^2 y^2 e^{x^3} dx \]
This can be rewritten as:
\[ 3x^2 y^2 e^{x^3} dx + y dx - x dy = 0 \]

### Step 2: Dividing by \( y^2 \)
Next, we divide the entire equation by \( y^2 \):
\[ 3x^2 e^{x^3} dx + \frac{y}{y^2} dx - \frac{x}{y^2} dy = 0 \]
This simplifies to:
\[ 3x^2 e^{x^3} dx + \frac{1}{y} dx - \frac{x}{y^2} dy = 0 \]

### Step 3: Rearranging Terms
Now, we can rearrange the terms:
\[ 3x^2 e^{x^3} dx = -\left(\frac{1}{y} dx - \frac{x}{y^2} dy\right) \]
This can be expressed as:
\[ 3x^2 e^{x^3} dx = -\left(\frac{dx}{y} - \frac{x dy}{y^2}\right) \]

### Step 4: Integrating Both Sides
Next, we integrate both sides. The left side requires substitution:
Let \( t = x^3 \), then \( dt = 3x^2 dx \) or \( dx = \frac{dt}{3x^2} \). Thus, we have:
\[ \int 3x^2 e^{x^3} dx = \int e^t dt = e^t + C = e^{x^3} + C \]

The right side can be rewritten as:
\[ \int \left(-\frac{dx}{y} + \frac{x dy}{y^2}\right) \]
This can be simplified to:
\[ -\int \frac{dx}{y} + \int \frac{x dy}{y^2} \]
The first integral is straightforward, while the second can be recognized as:
\[ \int d\left(\frac{x}{y}\right) \]

### Step 5: Combining Results
After integrating both sides, we have:
\[ e^{x^3} + C = \frac{x}{y} \]

### Step 6: Rearranging for the Final Solution
Rearranging gives us:
\[ x = y(e^{x^3} + C) \]

### Final Result
Thus, the solution to the differential equation is:
\[ x = y(e^{x^3} + C) \]