Q. 184.0( 3 Votes )

Solve the initial value problem: .

Answer :

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 - ey = t


On differentiating this, we get


ey 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 - ey) = 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 - ey) = log(x + 1) + 0


log(2 - ey) = - log(x + 1)







Taking log both side,




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


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