Find the general solution of the following :
`(4+5 sin x)(dy)/(dx)= cos x`.

Answer

To solve the differential equation \((4 + 5 \sin x) \frac{dy}{dx} = \cos x\), we will follow these steps:

### Step 1: Separate the variables
We can rewrite the equation to separate the variables \(y\) and \(x\):

\[ \frac{dy}{dx} = \frac{\cos x}{4 + 5 \sin x} \]

### Step 2: Integrate both sides
Now we will integrate both sides. The left side integrates to \(y\), and the right side requires integration of \(\frac{\cos x}{4 + 5 \sin x}\):

\[ \int dy = \int \frac{\cos x}{4 + 5 \sin x} dx \]

### Step 3: Use substitution for the right side
Let \(t = 4 + 5 \sin x\). Then, differentiate \(t\) with respect to \(x\):

\[ dt = 5 \cos x \, dx \quad \Rightarrow \quad dx = \frac{dt}{5 \cos x} \]

Now, substitute \(dx\) into the integral:

\[ \int \frac{\cos x}{t} \cdot \frac{dt}{5 \cos x} = \int \frac{1}{5t} dt \]

### Step 4: Integrate the right side
Now we can integrate:

\[ \int \frac{1}{5t} dt = \frac{1}{5} \ln |t| + C \]

### Step 5: Substitute back for \(t\)
Now substitute back \(t = 4 + 5 \sin x\):

\[ y = \frac{1}{5} \ln |4 + 5 \sin x| + C \]

### Final Solution
Thus, the general solution of the differential equation is:

\[ y = \frac{1}{5} \ln |4 + 5 \sin x| + C \]