what is the general solution of
` (1+e^(x)) y dy = e^(x) dx ` ?
where 'c' is a constant of integration

Options

A. `y^(2) =ln [c^(2) (e^(x) +1)^(2)]`
B. `y = ln[c (e^(x)+1)]`
C. `y^(2)=ln [c (e^(x) +1)]`
D. none of these

Answer

A

Explanation

The given differential equation is
` (1+e^(x)) y dy = e^(x) dx`
By separating the variable, we get
` ydy = (e^(x))/(1+e^(x)) dx`
Integrating on both the sides,
` Rightarrow int ydy = int (e^(x)/(1+e^(x))) dx`
` Rightarrow y^(2)/2 = log (1+e^(x)) + log c`
` Rightarrow y^(2)= 2log [c (1+e^(x))]`
( log m + log n = log mn )
` Rightarrow y^(2) = log [c^(2)(1+e^(x))^(2)]`