differential equation

• Nov 4th 2012, 12:36 AM
matmman
differential equation
Hello:

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

• Nov 4th 2012, 01:08 AM
chiro
Re: differential equation
Hey matmman.

Have you considered the Bernoulli solution?

Bernoulli differential equation - Wikipedia, the free encyclopedia
• Nov 4th 2012, 01:11 AM
Prove It
Re: differential equation
Quote:

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 \begin{align*} e^{-x} \end{align*} being subtracted...
• Nov 4th 2012, 01:12 AM
JJacquelin
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)
• Nov 4th 2012, 01:13 AM
Prove It
Re: differential equation
Wolfram gets this...
• Nov 4th 2012, 01:46 AM
JJacquelin
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).
• Nov 4th 2012, 02:45 AM
matmman
Re: differential equation
thanks for all

can we solve this equation numerically by matlab using ode or bvp4c function
• Nov 4th 2012, 09:29 PM
matmman
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
• Nov 5th 2012, 12:17 AM
JJacquelin
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) >>
• Nov 5th 2012, 02:35 AM
matmman
Re: differential equation
please can you send send the solution in detial
• Nov 5th 2012, 06:31 AM
JJacquelin
Re: differential equation
Quote:

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.