write integrating factor of the following differential equation:-
`(dx)/(dy)+x cosy = siny`

Answer

To find the integrating factor of the given differential equation, we will follow these steps:

1. **Identify the standard form of the linear differential equation**:
The given equation is \(\frac{dx}{dy} + x \cos y = \sin y\).
This can be compared to the standard form \(\frac{dx}{dy} + P(y)x = Q(y)\), where \(P(y) = \cos y\) and \(Q(y) = \sin y\).

2. **Determine the integrating factor**:
The integrating factor, \(\mu(y)\), is given by the formula:
\[ \mu(y) = e^{\int P(y) \, dy} \]
In our case, \(P(y) = \cos y\).

3. **Calculate the integral of \(P(y)\)**:
We need to compute the integral:
\[ \int \cos y \, dy \]
The integral of \(\cos y\) is \(\sin y\).

4. **Substitute the integral back into the integrating factor formula**:
Now we substitute the result of the integral into the formula for the integrating factor:
\[ \mu(y) = e^{\int \cos y \, dy} = e^{\sin y} \]

5. **Final result**:
Therefore, the integrating factor for the given differential equation is:
\[ \mu(y) = e^{\sin y} \]