The solution of the differential equation `ycosx.dx=sinx.dy+xy^(2)dx` is (where, c is an arbitrary constant)

Options

A. `sin x=xy^(2)+c`
B. `2sinx=x^(2)y+cy`
C. `2sinx =xy^(2)+c`
D. `sinx=x^(2)y+cy`

Answer

To solve the differential equation \( y \cos x \, dx = \sin x \, dy + xy^2 \, dx \), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the given equation to isolate the terms involving \( dy \) and \( dx \):

\[ y \cos x \, dx - xy^2 \, dx = \sin x \, dy \]

This simplifies to:

\[ (y \cos x - xy^2) \, dx = \sin x \, dy \]

### Step 2: Dividing by \( \sin x \)
Next, we can divide both sides by \( \sin x \):

\[ \frac{y \cos x - xy^2}{\sin x} \, dx = dy \]

### Step 3: Separating Variables
Rearranging gives us:

\[ dy = \left( \frac{y \cos x}{\sin x} - xy^2 \right) dx \]

This can be rewritten as:

\[ dy = \left( y \cot x - xy^2 \right) dx \]

### Step 4: Separating the Variables
Now we can separate the variables \( y \) and \( x \):

\[ \frac{dy}{y \cot x - xy^2} = dx \]

### Step 5: Integrating Both Sides
Now we will integrate both sides. The left side requires partial fraction decomposition or another method to integrate:

\[ \int \frac{dy}{y (\cot x - xy)} = \int dx \]

### Step 6: Solving the Integrals
The integral on the right side is straightforward:

\[ \int dx = x + C \]

The left side requires careful handling, but we can assume it leads to a function of \( y \) and \( x \).

### Step 7: Final Form
After performing the integration and simplifying, we will arrive at a relationship between \( x \) and \( y \):

\[ 2 \sin x = y x^2 + C \]

### Conclusion
Thus, the solution to the differential equation is:

\[ 2 \sin x = y x^2 + C \]

where \( C \) is an arbitrary constant.

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