1. ## differential equation

Hello:

I have to solve the following differential equation as shown in the attachment

2. ## Re: differential equation

Hey matmman.

Have you considered the Bernoulli solution?

Bernoulli differential equation - Wikipedia, the free encyclopedia

3. ## Re: differential equation

Originally Posted by chiro
Hey matmman.

Have you considered the Bernoulli solution?

Bernoulli differential equation - Wikipedia, the free encyclopedia
It's not Bernoulli because of the extra \displaystyle \displaystyle \begin{align*} e^{-x} \end{align*} being subtracted...

4. ## Re: differential equation

Hi !
it is a Riccati ODE.
First, change the variable : t=exp(-x)
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)

5. ## Re: differential equation

Wolfram gets this...

6. ## Re: differential equation

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).

7. ## Re: differential equation

thanks for all

can we solve this equation numerically by matlab using ode or bvp4c function

8. ## Re: differential equation

hi JJacquelin:
if we subsitute y by (1/4) (df/dt)(1/f) then how the function will be
what about dy/dt replace it by (d2f/dt2)(1/4f) and y^2 replace it by (df/dt)^2*(1/4)^2

9. ## Re: differential equation

Sorry, there was a typo :
y= 4 (df/dt)(1/f)
This was correctly written at the end of my post :
<< Then derivate it and bring f(t) and (df/dt) into y= 4(df/dt)*(1/f) >>

10. ## Re: differential equation

please can you send send the solution in detial

11. ## Re: differential equation

Originally Posted by matmman
please can you send send the solution in detial
I will not write the solution in whole details. When we reach the point where the Bessel functions appear, I suppose that the properties of these functions are known. Then, going into details would be a waste of time.