I developed a mathematics model which can be described by 2nd order equation with constant coefficients as shown below.
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + A\frac{\mathrm{d} y}{\mathrm{d} x}- B\ W(Cy) = 0$$
Where, $A$, $B$ and $C$ are real numbers, and $B>0, C>0$. $W(z)$ is the principle branch of Lambert W function, which gives $z=W(z)\exp[W(z)]$. For convenience, we just consider the solution in $x∈[0, 1], y>0$.
As you can see, this equation is in simple form, but unfortunately contains the Lambert W function. It indicates that the solution, at least, cannot be expressed with elementary functions.
I have tried a lot of ways, but I failed due to my limited mathematics knowledge. However, I still don't want to compromise to use the numerical method to solve it. Can we 'smartly' transform the ODE, so that we can 'smartly' get the explicit solution of $y=f(x)$ which contains e.g. Lambert W function?
I deeply appreciate any of your help and hint!
Best Answer
This answer doesn't solve the problem but intends to simplify the ODE. $$\frac{d^2y}{dx^2} + A\frac{dy}{dx}- B\ W(Cy) = 0$$ This is a second order ODE of autonomous kind. The usual way to reduce the order is the change of function : $$\frac{dy}{dx}=u(y)\quad\implies\quad \frac{d^2y}{dx^2}=\frac{du}{dy}\frac{dy}{dx}=u\frac{du}{dy}$$ $$u(y)\frac{du}{dy} + A\:u(y)- B\ W(C\,y) = 0$$ This is a first order nonlinear ODE.
Now one has to solve for $u(y)$ an ODE of lower order. But this doesn't help us very much because this is extremely unlikely to be possible with such a special function involved. Even with simpler functions $f(y)$ solving $u(y)\frac{du}{dy} + A\:u(y)+f(y) = 0$ leads to complicated solutions or is often impossible analytically due to the non linearity. For example the Abel's differential equations of the second kind.
In addition : How to get ride of the special function.
Let $\quad W(Cy)=t(y)$
$W(Cy)e^{W(Cy)}=Cy\quad\implies\quad te^{t}=Cy\quad\implies\quad C\,dy=e^t(t+1)dt$
$$\frac{du}{dy}=\frac{du}{dt}\frac{dt}{dy}=\frac{C}{e^t(t+1)}\frac{du}{dt}$$
$$u\frac{C}{e^t(t+1)}\frac{du}{dt} + A\,u- B\,t = 0$$
$$\boxed{u\,\frac{du}{dt} + \frac{1}{C}e^t(t+1)\left(A\,u- B\,t\right) = 0}$$ This is a first order nonlinear ODE to be solved for $u(t)$.
Unfortunately even without special function into it solving this ODE for solution in terms of a finite number of standard functions seems not possible in the general case of the parameters.