The general solution of differential equation `(e^(x)+1)ydy=(y+1)(e^(x))dx"is"`

Options

A. `(y+1)=k(e^(x)+1)`
B. `y+1=e^(x)+1+k`
C. `y=log {k(y+1)(e^(x)+1)}`
D. `y=log{(e^(x)+1)/(y+1)}+k`

Explanation

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

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ (e^{x}+1) y \, dy = (y+1) e^{x} \, dx \]
We can rearrange this to separate variables. Dividing both sides by \((y+1)e^{x}\) and multiplying both sides by \(dx\), we get:
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