Tommorow's too late (Posts about de)https://newkozlukov.gitlab.io/enContents © 2019 <a href="mailto:newkozlukov@gmail.com">Sergei Kozlukov</a> Fri, 13 Dec 2019 08:25:54 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rss- Variation of parametershttps://newkozlukov.gitlab.io/posts/variation/Sergei Kozlukov<p>
I've been wondering for a while, what kind of logic can lead to the method of
variation of parameters (i.e. it's trivial to prove, but why would one ever try
to find solution this way).
</p>
<p>
Now I found a convincing interpretation in the book "Обыкновенные
Дифференциальные Уравнения" (ODE) by
<a href="https://en.wikipedia.org/wiki/Vladimir_Arnold">Arnold</a>
This interpretation comes from the celestial mechanics and serves a basis
for all of the perturbation methods.
</p>
<p>
In this book the story for the variation of parameters method
begins with a sample model: the planets moving around the sun.
The first approximation is the motion according to Kepler's laws.
Then the objective is to count deviations,
caused by planets attracting each other.
And the idea is to suppose that planets would still follow Kepler's laws,
but the perturbation would make parameters vary over time.
</p>
<p>
The same idea applies to the inhomogeneous linear DE's. The solution is \(\phi =
\phi_h + \phi_p\) the sum of solution for homogeneous system and particular
solution for inhomogeneous one.
</p>
<p>
Let's suppose that \(\phi_h\) is kind of principal (undisturbed) part of the
solution, and \(\phi_p\) is some disturbance caused by inhomogenuity.
It could happen that disturbance can be described by variations of constants in
\(\phi_h\). Now let's just subtitute such solution into equation: voila!
That's the intuition that Zadorozhny's course lacked
</p>demathhttps://newkozlukov.gitlab.io/posts/variation/Tue, 14 Jun 2016 21:00:00 GMT