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

Explanation

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

### Step 1: Rearranging the Equation
We start with the equation:
\[ y \, dx - x \, dy + 3x^2 y^2 e^{x^3} \, dx = 0 \]
Rearranging gives us:
\[ y \, dx - x \, dy = -3x^2 y^2 e^{x^3} \, dx \]

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

### Step 3: Identifying the Left Side as a Derivative
The left-hand side can be recognized as the derivative of \( \frac{x}{y} \):
\[ d\left(\frac{x}{y}\right) = \frac{1}{y} \, dx - \frac{x}{y^2} \, dy \]
Thus, we can rewrite the equation as:
\[ d\left(\frac{x}{y}\right) = -3x^2 e^{x^3} \, dx \]

### Step 4: Integrating Both Sides
Now we integrate both sides:
\[ \int d\left(\frac{x}{y}\right) = \int -3x^2 e^{x^3} \, dx \]
The left side integrates to:
\[ \frac{x}{y} = \int -3x^2 e^{x^3} \, dx \]

### Step 5: Substituting for Integration
To solve the integral on the right, we can use substitution. Let:
\[ t = e^{x^3} \implies dt = 3x^2 e^{x^3} \, dx \implies dx = \frac{dt}{3x^2 e^{x^3}} \]
Thus, the integral becomes:
\[ \int -dt = -t + C \]
Substituting back gives:
\[ \frac{x}{y} = -e^{x^3} + C \]

### Step 6: Rearranging for the Final Solution
Rearranging the equation, we get:
\[ \frac{x}{y} + e^{x^3} = C \]
or equivalently:
\[ x + y e^{x^3} = C y \]

### Final Answer
The solution to the differential equation is:
\[ \frac{x}{y} + e^{x^3} = C \]