Solve the initial value problem: .

Given, A differential equation

To Find: Find the solution of the given differential equation

Explanation: We have given

Bifurcate the terms of y and x w.r.t to dy and dx respectively, we get

Now, Integrating Both Sides,

Let 2 - e^y = t

On differentiating this, we get

e^y dy = dt

Now, Substitute the value of t and dt , we get

- log t = log (x + 1) + C

putting back the value of t

- log(2 - e^y) = log(x + 1) + C …(1)

Now, The solution , when y(0) = 0

Then, put y = 0 and x = 0 in equation (i)

- log(2 - e⁰) = log(0 + 1) + C

- log(2 - 1) = log 1 + C

Since, log 1 = 0

- log 1 = 0 + C

- 0 = 0 + C

C = 0

On Putting the value of C in equation (i), we get

- log(2 - e^y) = log(x + 1) + 0

log(2 - e^y) = - log(x + 1)

Taking log both side,

Hence, This the required solution of the given differential equation.