What is the solution for (dx)/(dt)=(((e^tx+1))/(1-e^t)) ?

The solution for (dx)/(dt)=(((e^tx+1))/(1-e^t)) is x=(c_{1})/(1-e^t)+t/(1-e^t)

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(dx)/(dt)=(((e^tx+1))/(1-e^t))

\frac{d x}{d t} = (\frac{( e ^{t} x + 1 )}{1 - e ^{t}})

x = \frac{c _{1}}{1 - e ^{t}} + \frac{t}{1 - e ^{t}}

Solution steps

\frac{d x}{d t} = (\frac{( e ^{t} x + 1 )}{1 - e ^{t}})

Solve exact ODE:

x = \frac{c _{1}}{1 - e ^{t}} + \frac{t}{1 - e ^{t}}

x = \frac{c _{1}}{1 - e ^{t}} + \frac{t}{1 - e ^{t}}

What is the solution for (dx)/(dt)=(((e^tx+1))/(1-e^t)) ?
The solution for (dx)/(dt)=(((e^tx+1))/(1-e^t)) is x=(c_{1})/(1-e^t)+t/(1-e^t)