Hello:
I have to solve the following differential equation as shown in the attachment
please help me how to solve this differential equation
Hey matmman.
Have you considered the Bernoulli solution?
Bernoulli differential equation - Wikipedia, the free encyclopedia
Hi !
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).