If the solution of the differential equation `y^(3)x^(2)cos(x^(3))dx+sin(x^(3))y^(2)dy=(x)/(3)dx` is `2sin(x^(3))y^(k)=x^(2)+C` (where C is an arbitrary constant), then the value of k is equal to

Options

A. 3
B. 2
C. 1
D. 4

Answer

To solve the given differential equation and find the value of \( k \), we will follow these steps:

### Step 1: Write the given differential equation
The differential equation is given as:
\[ y^3 x^2 \cos(x^3) \, dx + \sin(x^3) y^2 \, dy = \frac{x}{3} \, dx \]

### Step 2: Rearranging the equation
We can rearrange the equation by moving everything to one side:
\[ y^3 x^2 \cos(x^3) \, dx + \sin(x^3) y^2 \, dy - \frac{x}{3} \, dx = 0 \]

### Step 3: Multiply through by 3
To simplify the equation, we can multiply through by 3:
\[ 3y^3 x^2 \cos(x^3) \, dx + 3\sin(x^3) y^2 \, dy - x \, dx = 0 \]

### Step 4: Identify the total differential
Notice that the left-hand side can be expressed as a total differential:
\[ d(\sin(x^3) y^3) = x \, dx \]
This means we can express the left side as:
\[ d(\sin(x^3) y^3) = x \, dx \]

### Step 5: Integrate both sides
Integrating both sides gives:
\[ \sin(x^3) y^3 = \frac{x^2}{2} + C \]
where \( C \) is the constant of integration.

### Step 6: Rearranging the equation
Now, we can rearrange this equation to isolate \( y^3 \):
\[ y^3 = \frac{x^2}{2\sin(x^3)} + \frac{C}{\sin(x^3)} \]

### Step 7: Express in the required form
The question states that the solution can be expressed as:
\[ 2\sin(x^3) y^k = x^2 + C \]
To match this with our integrated form, we can multiply both sides by 2:
\[ 2\sin(x^3) y^3 = x^2 + 2C \]
This means we can set \( k = 3 \).

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{3} \]