I have to solve the following differential equation as shown in the attachment
please help me how to solve this differential equation
Have you considered the Bernoulli solution?
Bernoulli differential equation - Wikipedia, the free encyclopedia
it is a Riccati ODE.
First, change the variable : t=exp(-x)
leading to dy/dt = -(y²/4)+(y/t)+1
Then let y= (1/4) (df/dt)(1/f) where f(t) is the new unknown function.
This leads to a linear second order ODE of the Bessel Kind.
Solve it for f(t) in termes of t*BeselI[1, t/2] and t*BesselK[1, t/2]
Then derivate it and bring f(t) and (df/dt) into y= 4(df/dt)*(1/f)
Finally, remplace t by exp(-x)
Wolfram gives the result that you will obtain thanks to the method which was given in my preceeding post.
Note that t=sqrt(exp(-2x)). In the WolframAlpha formula, the denominator of the fraction is the function f(t) and the numerator is 4(df/dt).