Analytical solution of nonlinear ordinary differential equation

Question

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).

Answer 1

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.