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 can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ y \cos x \, dx - \sin x \, dy = xy^2 \, dx \]
This can be rewritten as:
\[ y \cos x \, dx - \sin x \, dy - xy^2 \, dx = 0 \]
Combining the \( dx \) terms gives us:
\[ (y \cos x - xy^2) \, dx - \sin x \, dy = 0 \]

### Step 2: Dividing by \( y^2 \)
Next, we divide the entire equation by \( y^2 \):
\[ \frac{y \cos x}{y^2} \, dx - \frac{\sin x}{y^2} \, dy - x \, dx = 0 \]
This simplifies to:
\[ \frac{\cos x}{y} \, dx - \frac{\sin x}{y^2} \, dy - x \, dx = 0 \]

### Step 3: Rearranging Again
Rearranging gives us:
\[ \frac{\cos x}{y} \, dx - x \, dx = \frac{\sin x}{y^2} \, dy \]
Factoring out \( dx \):
\[ \left( \frac{\cos x - xy}{y} \right) \, dx = \frac{\sin x}{y^2} \, dy \]

### Step 4: Using the Method of Separation of Variables
Now we can separate the variables:
\[ \frac{y^2}{\sin x} \, dy = \frac{y}{\cos x - xy} \, dx \]

### Step 5: Integrating Both Sides
Integrate both sides:
\[ \int \frac{y^2}{\sin x} \, dy = \int \frac{y}{\cos x - xy} \, dx \]
The left side integrates to:
\[ \frac{y^3}{3 \sin x} + C_1 \]
The right side requires a substitution or integration technique.

### Step 6: Final Expression
After performing the integration and simplifying, we arrive at:
\[ \sin x = y \left( \frac{x^2}{2} + C \right) \]
Where \( C \) is an arbitrary constant.

### Step 7: Final Answer
Thus, the solution of the differential equation is:
\[ 2 \sin x = x^2 y + C y \]