`(e^(y)+1)cos x dx +e^(y) sin x dy=0`. The solution of this diff. eqn. is `cos x(e^(y)+1)=c`.

Answer

To solve the differential equation \((e^{y}+1)\cos x \, dx + e^{y} \sin x \, dy = 0\), we can rearrange it into a more manageable form.

### Step 1: Rearranging the Equation
We start by rewriting the equation:
\[ (e^{y}+1)\cos x \, dx + e^{y} \sin x \, dy = 0 \]
This can be rearranged as:
\[ e^{y} \sin x \, dy = - (e^{y}+1) \cos x \, dx \]

### Step 2: Separating Variables
Next, we separate the variables \(y\) and \(x\):
\[ \frac{dy}{(e^{y}+1)} = -\frac{\cos x}{e^{y} \sin x} \, dx \]

### Step 3: Integrating Both Sides
Now, we integrate both sides. The left side will be integrated with respect to \(y\) and the right side with respect to \(x\):
\[ \int \frac{dy}{(e^{y}+1)} = -\int \frac{\cos x}{e^{y} \sin x} \, dx \]

### Step 4: Solving the Integrals
The integral on the left side can be solved using a substitution or recognizing it as a standard integral. The right side can also be simplified:
\[ \int \frac{dy}{(e^{y}+1)} \quad \text{and} \quad -\int \frac{\cos x}{\sin x} \, dx \]
The right side simplifies to \(-\ln|\sin x|\).

### Step 5: Combining the Results
After integrating, we combine the results:
\[ \ln(e^{y}+1) = -\ln|\sin x| + C \]
Exponentiating both sides gives:
\[ e^{y}+1 = \frac{C}{|\sin x|} \]

### Step 6: Final Form
We can rearrange this to get the final implicit solution:
\[ \cos x(e^{y}+1) = C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \cos x(e^{y}+1) = C \]