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 equation:
\[ x dy - y dx + 3x^2 y^2 e^{x^3} dx = 0 \]
This can be rewritten as:
\[ 3x^2 y^2 e^{x^3} dx = y dx - x dy \]

### Step 2: Dividing by \( y^2 \)
Next, we divide both sides by \( y^2 \):
\[ 3x^2 e^{x^3} dx = \frac{y dx - x dy}{y^2} \]

### Step 3: Recognizing the Derivative
The right-hand side can be recognized as the derivative of \( \frac{x}{y} \):
\[ \frac{y dx - x dy}{y^2} = d\left(\frac{x}{y}\right) \]
Thus, we can rewrite the equation as:
\[ 3x^2 e^{x^3} dx = d\left(\frac{x}{y}\right) \]

### Step 4: Integrating Both Sides
Now we integrate both sides:
\[ \int 3x^2 e^{x^3} dx = \int d\left(\frac{x}{y}\right) \]
To solve the left integral, we use the substitution \( t = x^3 \), which gives \( dt = 3x^2 dx \). Thus, we have:
\[ \int e^t dt = e^t + C = e^{x^3} + C \]
So, we have:
\[ e^{x^3} = \frac{x}{y} + C \]

### Step 5: Rearranging the Result
Rearranging gives us:
\[ y e^{x^3} = x + Cy \]
This can be simplified to:
\[ y e^{x^3} - Cy = x \]
Factoring out \( y \):
\[ y(e^{x^3} - C) = x \]

### Final Solution
Thus, the solution of the differential equation is:
\[ y = \frac{x}{e^{x^3} - C} \]