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.
Any help will be highly helpful.
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).
Best Answer
If you can assume that $\gamma$ is a small parameter, then write
$$x(t) = x_0(t) + \gamma x_1(t) + \gamma^2 x_2(t) + \cdots$$ $$y(t) = y_0(t) + \gamma y_1(t) + \gamma^2 y_2(t) + \cdots$$
i.e., assume such convergent series exist. A zeroth order solution is $y_0(t)=y(0)=y_0$, a constant, and the equation for $x_0(t)$ becomes
$$\frac{dx_0}{dt} = 2 (W+y_0) x_0-4 x_0^3$$
This equation is integrable:
$$\int \frac{dx_0}{2 (W+y_0) x_0-4 x_0^3} = t \implies \frac{1}{4 (W + y_0)} \log{\left (\frac{x_0(t)}{2 x_0(t)^2-W-y_0}\right)} = t+C$$
Solve for $x_0(t)$, then plug into the first order equation for $y_1(t)$:
$$\gamma \frac{dy_1}{dt} = \gamma(x_0(t)^2-y_0) \implies \frac{dy_1}{dt} = x_0(t)^2-y_0$$
Integrate with respect to $t$ to get $y_1(t)$, then plug into $x$ equation:
$$\frac{d}{dt} (x_0+\gamma x_1) = 2 (W + y_0+\gamma y_1) (x_0+\gamma x_1) - 4 (x_0+\gamma x_1)^3$$
Note that $(x_0+\gamma x_1)^3 = x_0^3 + 3 \gamma x_0^2 x_1 + O(\gamma^2)$. Coefficient of $\gamma^0$ is zero because of above equation. Equating coefficients of $\gamma^1$, we get
$$\frac{d x_1}{dt} = 2 (W+y_0) x_1 + x_0 y_1 - 12 x_0^2 x_1$$
This is an inhomogeneous 1st order equation for $x_1$, which may be solved using known techiques (e.g., integration factor).
At this point, you may repeat this process to get higher powers of $\gamma$. I do not have a proof that the resulting series converges.