The hidden secrets

It was long time ago, when I interested in natural sciences. I was about six years old and asked my dad to explain me what are the stars on the sky and what is Jupiter. The following night my dad waked me up, and showed me the night sky, and since then I have had a dream to understand what kind of universe is and how it evolves; how stars first born and shine until they finally die after many many - almost countless - years. And how one can ever know where and when asteroids next go on their orbit, when the Sun rises? To be able to answer these questions was my dream before I was able to read myself.

Now, about 20 years older, I should know a lot more than six years old child. But it still feels that I know simply nothing about the universe. And when I think I know something, the reality surprises me. It shows how much more beautiful and complex the world really is and how it can exceed imagination. In this blog I try to understand the world better and share some parts with you. I hope I have some of the enthusiasm of six years old me when I step on this path.

The main topics of the blog are usually related to mathematics, physics and astronomy, but who'd know if there was something stranger occasionally. It should also be noted that some things I post here, can be difficult to understand, but I've found them so interesting that I like to write about them. In addition, almost in all topics, I can say that there is use for math skills when reading!

As for an easy example of the topics of this blog, I took an ordinary differential equation (ode):

\[y' = 1 - (y - x)^2\]

for some short analyze. Let's make this an initial value problem by setting $y(0) = -1$. And now I ask, what is the value $y(2)$? (These kind of problems are very usual in physics so even though this problem is not trying to model anything, it hopefully show us something about the nature of such kind of mathematics.)

To answer to the question, the ode should be solved. It is nonlinear and so there are not any method to solve it directly. (Thought, there is a fairly easy procedure to solve it analytically but let's talk about it a little later.) Usually, when one wants to see the solution and can't or doesn't want to solve the equation analytically, numerical methods are used. There are several of them, and if anyone is interested to test how a specific method works in case of this ode, feel free to try and see. I chose an explicit Euler method (step size $h = 0.1$) and got something like $y(2) \approx -5 \cdot 10^{103}$. Looks weird as the value changes so much, and it hints that there happens something we maybe can't directly see by looking at the ode.

The next approach for 'unsolvable' differential equations is usually write the solution to a somekind of serie. Power series are often used, and I can write (starting at $x=0$):

\[y(x) = -1 - \sum_{v=2}^{\infty}x^v.  \]

And now we can see, that value $y(2)$ can't find by using this serie as the value of the sum goes to infinity. But what it means? Does it mean that there is no value $y(2)$? Well, that might be one case, but not only one. And without further analysis there is not much to say. In this case the ode can be solved analyticly (substitute $y(x) = x + p(x)$ and solve what is rest):

\[y(x) = x + \frac{1}{x-1}.\]

And now we see that the solution of the ode we were looking for, is discontinuous at $x=1$ and this explains the 'odd' behaviour. But without this analytical solution it would have been very hard to find that $y(2) = 3$. And in the future problems in this blog the analytical solutions are not usually known - or they are complicated.

Numerical and serie solution fail at this time. (Click to see larger image.)