Analytical solution of nonlinear ordinary differential equation

[nitin7785]

Dear All,

I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative solution.

$$\frac{dx}{dt} = 2Wx + 2xy - 4x^{3}$$$$\frac{dy}{dt} = \gamma \, (x^{2} - y)$$

Kindly help me with any methods you that might work and it will be great if you can provide few references where i can read about those methods.

Also If somebody can help me about how I can use fixed point analytic method to solve this differential equations and some references on it, will be very useful too.

Thanks a lot in advance.

PS. I tried homotopy perturbation analysis and simple iteration procedure to try to solve it and it diverges after some time(good only for early short times).

 

[GregoryS]

The most common method to solve such equations you may find there: http://eqworld.ipmnet.ru/en/solutions/sysode/sode0303.pdf

 

[Chestermiller]

I would multiply the first equation by x, then reexpress the equations in terms of z, where z = x². I would also solve for the long time solution, which is z = y = W. How do W and γ compare in magnitude? The answer to this question is important. Also, consider the behavior at long times, less than t = ∞, by writing z = W+ε_z and y = W+ε_y. What are the initial conditions on the problem? This could also be relevant.

Chet

 

[nitin7785]

I would multiply the first equation by x, then reexpress the equations in terms of z, where z = x². I would also solve for the long time solution, which is z = y = W. How do W and γ compare in magnitude? The answer to this question is important. Also, consider the behavior at long times, less than t = ∞, by writing z = W+ε_z and y = W+ε_y. What are the initial conditions on the problem? This could also be relevant.

Chet

Thanks for your answer.

W is between 0 and 1.

γ is between 0 to ∞.

Initial conditions are close to zero but cannot be zero.

It will be very kind of you if you can elaborate on your answer.

Thanks,
nitin

 

[Chestermiller]

Thanks for your answer.

W is between 0 and 1.

γ is between 0 to ∞.

Initial conditions are close to zero but cannot be zero.

It will be very kind of you if you can elaborate on your answer.

Thanks,
nitin

What is the ratio of W to γ in typical situations? This ratio controls everything in the solution trajectory. Have you done what I suggested about replacing x by z? You can also reduce the equations to dimensionless form, at least in terms of the dependent variables, by setting z = Wz* and y = Wy* and solving for z* and y*. This will show that the time constant for the first equation is 1/W, and the time constant for the second equation is 1/γ. Please make the substitutions I have suggested, and show us what you got.

Chet

 

[epenguin]

:shy: I think it is usually worth trying to differentiate such equations and hope to be able to eliminate y and y', then get an equation in only x, x' and x'' which may be solvable analytically. It seems to me you get here a second order linear (not constant coefficients) which you can then solve reducing it to a first-order eq. in variables x and p (= x'), ridding it of second derivative by x'' = p.dp/dx.
This idea is explained in para. 70 of Piaggio's DE book (switching around notation) but I think I've just said it. It is rather lengthy calculation by my standards and I have not completed a calculation, so am not yet confident.

I have not yet been able to make out whether this

The most common method to solve such equations you may find there: http://eqworld.ipmnet.ru/en/solutions/sysode/sode0303.pdf

is the same method or a better or more general one and its rationale and I too would be grateful for some further elaboration. The quoted text reads like a confusing mistranslation - should 'is' be 'be'?