I'm trying to solve the following forth-order ODE with the shooting method:
$$\frac{1}{5}(y-2xy^\prime)=\frac{1}{x}\left\{\frac{xy^\prime}{y}+xy^3 \left[\frac{(xy^\prime)^\prime}{x} \right]^\prime \right\}^\prime$$ where $\prime$ denotes differentiation. It is clear that this ODE need 4 boundary conditions (BCs): two of which are given $y^\prime(0)=y^{\prime\prime\prime}(0)=0$, however, the other two values of $y(0)$ and $y^{\prime\prime}(0)$ are determined by shooting for BCs at infinity ($x_\text{Max}$).
It has been found that $y=Ax^{1/2}$ is a far-field asymptotic solution to the ODE by neglecting the nonlinear terms. Now, let me refer to the solutions with $y\sim x^{1/2}$ asymptotic behavior as the $x^{1/2}$ solutions. Here, I want to search just for this $x^{1/2}$ solutions by shooting method in which the ODE is integrated with a 4th-order R-K scheme from the origin to a certain end-value of $x_{max}$. In practice, I truncate the domain to $x_0<x<x_\text{Max}$ to avoid the singularity at $x=0$, and I would like to work with $x_0=10^{-4}$ and $x_\text{Max}=10$.
The shooting parameters $y(0)$ and $y^{\prime\prime}(0)$ will be adjusted so that $y-2xy^\prime=0$ or $y+4x^2y^{\prime\prime}=0$ at $x=x_\text{Max}$. The two BCs were chosen to be independent to involve low-order derivatives and to require $y\propto x^{1/2}$ at the end of the integration interval.
My questions are:
(1) How can I use a Taylor series (say, five-terms) to start the integration at $x_0=10^{-4}$. I am thinking the series should be located in the position of initial conditions, but I can't figure it out;
(2) How do I implement this searching method: firstly, find the solutions that satisfying the 1st BCs $y-2xy^\prime=0$ at $x_\text{Max}$, next, find the solutions that satisfying the 2nd BC $y+4x^2y^{\prime\prime}=0$ at $x_\text{Max}$. Then ParametricPlot
the shooting parameters $(y(0), y^{\prime\prime}(0))$ that corresponding to solutions satisfying the above-mentioned BCs respectively, thus the intersections of the two sets of curves correspond to the desired $x^{1/2}$ solutions.
(3) Is it possible to use Do
loop and FindRoot
to find the parameters, as shown in this answer?
Thanks!