The solution of the differential equation `sinye^(x)dx-e^(x)cos ydy=sin^(2)ydx` is (where, c is an arbitrary constant)

Options

A. `e^(x)siny=x+c`
B. `e^(x)=(x+c)siny`
C. `e^(x).x=siny+c`
D. `e^(x).siny=x^(2)+c`

Answer

To solve the differential equation given by:

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

we can rearrange the equation as follows:

1. **Rearranging the Equation**:
\[ \sin y e^x \, dx - e^x \cos y \, dy - \sin^2 y \, dx = 0 \]
This can be rewritten as:
\[ (\sin y e^x - \sin^2 y) \, dx - e^x \cos y \, dy = 0 \]

2. **Dividing by \(\sin^2 y\)**:
We can divide the entire equation by \(\sin^2 y\) to simplify it:
\[ \frac{\sin y e^x - \sin^2 y}{\sin^2 y} \, dx - \frac{e^x \cos y}{\sin^2 y} \, dy = 0 \]
This gives:
\[ \left(\frac{e^x}{\sin y} - 1\right) \, dx - \frac{e^x \cos y}{\sin^2 y} \, dy = 0 \]

3. **Identifying the Total Differential**:
Notice that we can express the left-hand side as a total differential:
\[ d\left(\frac{e^x}{\sin y}\right) = \frac{e^x}{\sin y} \, dx - \frac{e^x \cos y}{\sin^2 y} \, dy \]
Therefore, we can write:
\[ d\left(\frac{e^x}{\sin y}\right) = 0 \]

4. **Integrating**:
Integrating both sides gives:
\[ \frac{e^x}{\sin y} = C \]
where \(C\) is an arbitrary constant.

5. **Final Solution**:
Rearranging gives us the final solution:
\[ e^x = C \sin y \]