The solution of the differential equation :
`(dy)/(dx)=e^(x)+1` is :

Options

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

Answer

To solve the differential equation \(\frac{dy}{dx} = e^x + 1\), we will follow these steps:

### Step 1: Separate the variables
We can rewrite the equation as:
\[ dy = (e^x + 1) \, dx \]

### Step 2: Integrate both sides
Now, we will integrate both sides. The left side integrates to \(y\), and the right side requires us to integrate \(e^x\) and \(1\):
\[ \int dy = \int (e^x + 1) \, dx \]

### Step 3: Perform the integration
The integral of \(dy\) is simply \(y\). The integral of \(e^x\) is \(e^x\), and the integral of \(1\) is \(x\):
\[ y = e^x + x + C \]
where \(C\) is the constant of integration.

### Final Solution
Thus, the solution to the differential equation is:
\[ y = e^x + x + C \]

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