Dear All,
Here is question/problem. For y=y(x) and real x, solve the following ODE: y'-y^2+g(x)=0, where g(x) is a given positive function. For instance, g(x)=2e^x/(e^x+1)^2.
Any suggestions about how to find an explicit solution?
Many thanks,
Ciprian
Ricatti equation with coefficients q0(x) = -g(x), q1(x)=0, q2(x)=-1.
See link:
https://en.wikipedia.org/wiki/Riccati_equation
Matlab can solve similar differential equations, but most probably a solution does not exist for any function g(x). Try something like the following in Matlab
syms y(t)
ode = diff(y,t) - y^2 + 2*exp(t) == 0;
cond = y(0) == 0;
ySol(t) = dsolve(ode,cond)
I also added an initial condition. The above provides a solution for g(t)=2*exp(t). But if you try your suggestion g(t)=2*exp(t)/(exp(t)+1)^2 then Matlab cannot find a solution. You get: Warning: Unable to find explicit solution.
Numerical methods for solving differential equations can be used. Euler's method, Improved Euler's method, etc
Finite element method, Finite difference method can also be used.
Does it help to notice that g is a multiple of sech^2(x/2)? Also, the derivative of tanh(x/2) is a mutiple of sech^2(x/2) and tanh^2(x/2) is related to sech^(x/2).
This could help since the ODE above is in fact related to the 2nd-order ODE z''=g(x)z. The latter can be transformed by means of a carefully selected Ansatz to the 1st-order Ricatti equation above (with z solving another equation in terms of y only).
Many thanks,
Ciprian
It is possible to get a solution (may be not interesting for the case) in Octave
https://octave-online.net/
with
syms y(t) ode = diff(y,t) - y^2 + 2*exp(t) == 0; cond = y(0) == 0; ySol(t) = dsolve(ode,cond)
which is
y(t) == -2*t - pow(t, 2) + pow(t, 3) + (11.0/12.0)*pow(t, 4) - 37.0/60.0*pow(t, 5) + O(t**6)
Can you comment about it?
》
You might like to examine the real solvability of the problem first.
Efforts for any unphysical solution corresponding to a physical problem should be immediately discarded.
The best way is to check the global integrability detection, Painleve analysis, etc.
The details may be found at:
[1] A. C. Newell, “Perturbative theory of solitons,” Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1985).
[2] M. Lakshamanan and S. Rajasekhar: Nonlinear Dynamics: Integrability, Chaos, and Patterns (Springer, New York, 2002).
[3] H. Berestycki and Y. Porneau (eds), Nonlinear PDE's in Condensed Matter and Reactive Flows (Springer, New York, 2002).
Dear Ciprian,
I think you can you can use Mathematica or Maple to solve the equation. Here, I have provided the solution given by Maple. You can test it and see the results yourself.
Good Luck,
Mohsen
Point collocation methods can also be used for approximate solutions to non linear differential equations.
Dear Ciprian G. Gal ,
As you mentioned, the original differential equation is a Riccati ODE, and therefore (through a convenient change of variables) can be written as
u''-g(x)u=0
Can't really think on explicit solutions for it when g'(x)\neq 0... Nevertheless, work has been done on solving it through power series and special functions. Maybe checking the paper of Shivakumar and Zhang (2013) [1], and references therein, can help you find the solution that best suits your problem. Let me know how it goes :-)
Best wishes,
Sebastián
[1] Article On series solutions of and applications
Power series methods can be used to obtain approximate solutions to the problem.
Or s thé bernouilli ricatti équations.you have to détermine à particular solution and make chgt
Dear @ Ciprian G. Gal Hi,
As seems your equation is well-known Bernoulli equation in the differential equations context. So, the analytical solution for any given function g(x) can be obtained. First you should be use appropriate "change of variable" then solve obtained linear first order differential equation. u(x)=(y(x))^(1-n), n=2.
thank you
Dear Kamran Soltani ,
The equation is slightly different from a Bernoulli's in shape but radically different in complexity. As the linear term "y" is missing, the equation can't be solved with that change of variable. Anyway, we could use a different one, as suggested in previous answers, to write the Ricatti equation as a second-order ODE, which solutions are expressed in terms of power-series or special functions for any case where function g is not a constant [1]
Best wishes,
Sebastian
[1] Article On series solutions of and applications
Dear @
Sebastián Contreras
Yes you right I looked at the equation in a hurry (I thought the equation is in form y'-y^2*g(x)=0 ). But for that equation, it is Riccati differential equation, so for achieving to a closed form solution we there are two general way: 1) Reduction to a second linear order differential equation and 2) solution using the "change of variable" y=y1+(u)^(-1), where y1, is a particular solution of the equation. However, here, since there is general function (g(x)) in the first method the obtained linear second order equation is non-constant coefficient, therefore its solution cannot be obtained analytically unless it be a known equation. In the second method but, the problem of finding a particular solution, it's just as hard. Accordingly, a closed form solution only possible in very limited cases (in terms of g(x)). Finally, I think familiar numerical method such as 4th-order Runge-Kutta method or ... are more efficient.
Best wishes, Kamran Soltani
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