Solution of the differential y' = `(3yx^(2))/(x^(3)+2y^(4))` is

Options

A. `x^(3)y^(-1)=(2)/(3)y^(3)+c`
B. `x^(2)y^(-1)=(2)/(3)y^(3)+c`
C. `xy^(-1)=(2)/(3)y^(3)+c`
D. None of these

Explanation

To solve the differential equation \( y' = \frac{3yx^2}{x^3 + 2y^4} \), we will follow the steps outlined in the video transcript.

### Step-by-Step Solution:

1. **Rewrite the Differential Equation:**
We start with the given equation:
\[ \frac{dy}{dx} = \frac{3yx^2}{x^3 + 2y^4} \]

2. **Separate Variables:**
We can rearrange the equation to separate the variables \( y \) and \( x \):
\[ (x^3 + 2y^4) dy = 3yx^2 dx \]

3. **Rearrange the Terms:**
Rearranging gives us:
\[ x^3 dy + 2y^4 dy = 3yx^2 dx \]
Now, we can express it as:
\[ 2y^4 dy = 3yx^2 dx - x^3 dy \]

4. **Factor Out Common Terms:**
We can factor out \( y \) from the right-hand side:
\[ 2y^4 dy = y(3x^2 dx - x^3 dy) \]

5. **Divide by \( y^2 \):**
Dividing both sides by \( y^2 \) (assuming \( y \neq 0 \)):
\[ 2y^2 dy = \frac{3x^2 dx - x^3 dy}{y^2} \]

6. **Integrate Both Sides:**
Now we can integrate both sides:
\[ \int 2y^2 dy = \int \left(\frac{3x^2}{y^2} dx - \frac{x^3}{y^2} dy\right) \]
The left side integrates to:
\[ \frac{2}{3} y^3 \]
The right side can be simplified to:
\[ \frac{x^3}{y} + C \]

7. **Final Solution:**
Thus, we have:
\[ \frac{2}{3} y^3 = \frac{x^3}{y} + C \]
Rearranging gives us the implicit solution of the differential equation.

### Final Answer:
The solution of the differential equation is:
\[ \frac{2}{3} y^3 = \frac{x^3}{y} + C \]