Thread: differential equation

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

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 being subtracted...

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 gets this...

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).

thanks for all

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

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

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) >>

please can you send send the solution in detial

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.