## Variation of Parameters and More

Well, the last time I posted a project update, I had resigned myself to writing a constant simplifying function and putting the Constant class on the shelf. Well, just as I suspected, it was hell writing it, but I eventually got it working. Already, what I have in dsolve() benefits from it. I had many solutions with things like $\frac{x}{C_1}$ or $-C_1x$ in them, and they are now automatically reduced to just $C_1x$. Of course, the disadvantage to this, as I mentioned in the other post, is that it will only simplify once. Also, I wrote the function very specifically for expressions returned by dsolve. It only works, for example, with constants named sequentially like C1, C2, C3 and so on. Even with making it specialized, it was still hell to write. I was also able to get it to renumber the constants, so something like C2*sin(x) + C1*cos(x) would get transfered to C1*sin(x) + C2*cos(x). It uses Basic._compare_pretty() (thanks to Andy for that tip), so it will always number the constants in the order they are printed.

Once I got that working, it was just little work to finish up what I had already started with solving general linear homogeneous odes ($a_ny^{(n)} + a_{n-1}y^{(n-1)} + \dots + a_2y'' + a_1y' + a_0y = 0$ with $a_i$ constant for all $i$). Solving these equations is easy. You just set up a polynomial of the form $a_nm^n + a_{n-1}m^{n-1} + \cdots + a_2m^2 + a_1m + a_0 = 0$ and find the roots of it. Then you plug the roots into an exponential times $x^i$ for i from 1 to the multiplicity of the root (as in $Cx^ie^{root \cdot x}$). You usually expand the real and complex parts of the root using Euler’s Formula, and, once you simplify the constants, you get something like $x^ie^{realpart \cdot x}(C_1\sin{(impart \cdot x)} + C_2\cos{(impart \cdot x)})$ for each i from 1 to the multiplicity of the root. Anyway, with the new constantsimp() routine, I was able to set this whole thing up as one step, because if the imaginary part is 0, then the two constants will be simplified into each other. Also, SymPy has some good polynomial solving, so I didn’t have any problems there. I even made good use of the collect() function to factor out common terms, so you get something like $(C_1 + C_2x)e^{x}$ instead of $C_1e^{x} + C_2xe^{x}$, which for larger order solutions, can make the solution much easier to read (compare for example, $((C_1 + C_2x)\sin{x} + (C_3 + C_4x)\cos{x})e^{x}$ with the expanded form, $C_1e^{x}\sin{x} + C_2xe^{x}\sin{x} + C_3\cos{x}e^{x} + C_4x\cos{x}{e^x}$ as the solution to ${\frac {d^{4}}{d{x}^{4}}}f \left( x \right) -4\,{\frac {d^{3}}{d{x}^{3}}}f \left( x \right) +8\,{\frac {d^{2}}{d{x}^{2}}}f \left( x \right) -8\,{\frac {d}{dx}}f \left( x \right) +4\,f \left( x \right) =0$).

I entered all 30 examples from the relevant chapter of my text (Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard), and the whole thing runs in under 2 seconds on my machine. So it is fast, though that is mostly due to fast polynomial solving in SymPy.

So once I got that working well, I started implementing variation of parameters, which is a general method for solving all equations of form $a_ny^{(n)} + a_{n-1}y^{(n-1)} + \dots + a_2y'' + a_1y' + a_0y = F(x)$. The method will set up an integral to represent the particular solution to any equation of this form, assuming that you have all $n$ linearly independent solutions to the homogeneous equation $a_ny^{(n)} + a_{n-1}y^{(n-1)} + \dots + a_2y'' + a_1y' + a_0y = 0$. The coefficients $a_i$ do not even have to be constant for this method to work, although they do have to be in my implantation because otherwise it will not be able to find general solution to the homogeneous equation.

So, aside from doing my GSoC project this summer, I am also learning Linear Algebra, because I could not fit it in to my schedule next semester and I need to know it for my Knot Theory class. It turns out that it was very useful in learning the method of variation of parameters. I will explain how the method works below, but first I have a little rant.

Why is the Wikipedia article on variation of parameters the only website anywhere that covers variation of parameters in the general case? Every other site that I could find only covers 2nd order equations, which I understand is what is taught in most courses because applying it to anything higher can be tedious and deriving the nth order case requires knowledge of Cramer’s Rule, which many students may not know. But you would think that there would at least be sites that discuss what I am about to discuss below, namely, applying it to the general case of an nth order inhomogeneous linear ode. Even the Wolphram MathWorld article only explains the derivation for a second order linear ODE, mentioning at the bottom that it can be applied to nth order linear ODEs. I did find a website called Planet Math that covers the general case, but it wasn’t on the top of the Google results list and took some digging to find. It also has problems of its own, like being on a very slow server and some of the LaTeX on the page not rendering among them.

This partially annoys me because the Wikipedia article is not very well written. You have to read through it several times to understand the derivation (I will try to be better below). The Planet Math site is a little better, but like I said, it took some digging to find, and I actually found it after I had written up half of this post already.

But it is also part of a larger attitude that I am finding more and more of where anything that is not likely to be directly applied is not worth knowing and thus not worth teaching. Sure, it is not likely that any person doing hand calculations will ever attempt variation of parameters on an ode of order higher than 2 or 3, but that is what computer algebra systems like SymPy are for. Unfortunately, it seems that they are also in a large part for allowing you to not know how or why something mathematically is true. What difference does it make if variation of parameters can be applied to a 5th order ODE if I have to use Maple to do actually do it anyway. As long as the makers of Maple know how to apply variation of parameters to a nth order ODE, I can get along just fine. At least with SymPy, the source is freely available, so anyone who does desire to know how things are working can easily see. Anyway, I am done ranting now, so if you were skipping that part, this would be the point to start reading again.

So you have your linear inhomogeneous ODE: $a_ny^{(n)} + a_{n-1}y^{(n-1)} + \dots + a_2y'' + a_1y' + a_0y = F(x)$. $a_n$ cannot be zero (otherwise it would be a n-1 order ODE), so we can and should divide through by it. Lets pretend that we already did that, and just use the same letters. Also, I will rewrite $a_n$ as $a_n(x)$ to emphasize that the coefficients do not have to be constants for this to work. So you have your linear inhomogeneous ODE: $y^{(n)} + a_{n-1}(x)y^{(n-1)} + \dots + a_2(x)y'' + a_1(x)y' + a_0(x)y = F(x)$. So, as I mentioned above, we need n linearly independent solutions to the homogeneous equation $y^{(n)} + a_{n-1}(x)y^{(n-1)} + \dots + a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ to use this method. Let us call those solutions $y_1(x), y_2(x), \dots, y_n(x)$. Now let us write our particular solution as $y_p(x) = c_1(x)y_1(x) + c_2(x)y_2(x) + \dots + c_n(x)y_n(x)$. Now, if we substitute our particular solution in to the left hand side of our ODE, we should get $F(x)$ back. So we have $(y_p)^{(n)} + a_{n-1}(x)(y_p)^{(n-1)} + \dots + a_2(x)y_p'' + a_1(x)y_p' + a_0(x)y_p =$ $F(x)$. Now, let me rewrite $y_p$ as a summation to help keep things from getting too messy. I am also going to write $c_i$ instead of $c_i(x)$ on terms for additional sanity. Every variable is a function of x. $y_p(x) = \sum_{i=1}^{n} c_i y_i$. The particular solution should satisfy the condition of the ODE, so
$y_p^{(n)} + a_{n-1}y_p^{(n-1)} + \dots + a_2y_p'' + a_1y_p' + a_0y_p = F(x)$.

$(\sum_{i=1}^{n} c_i y_i)^{(n)} + a_{n-1}(\sum_{i=1}^{n} c_i y_i)^{(n-1)} + \dots + a_2(\sum_{i=1}^{n} c_i y_i)^{(2)} +$
$a_1(\sum_{i=1}^{n} c_i y_i)^{(1)} + a_0\sum_{i=1}^{n} c_i y_i = F(x)$.

Now, if we apply the product rule to this, things will get ugly really fast, because we have to apply the product rule on each term as many times as the order of that term (the first term would have to be applied n times, the second, n-1 times, and so on). But there is a trick that we can use. In the homogeneous case, there is no particular solution, so in that case the $c_i$ terms must all vanish identically because the solutions are linearly independent of one another. Thus, if we plug the particular solution into the homogeneous case, we get

$(\sum_{i=1}^{n} c_i y_i)^{(n)} + a_{n-1}(\sum_{i=1}^{n} c_i y_i)^{(n-1)} + \dots + a_2(\sum_{i=1}^{n} c_i y_i)^{(2)} +$
$a_1(\sum_{i=1}^{n} c_i y_i)^{(1)} + a_0\sum_{i=1}^{n} c_i y_i = 0$.

We already know that if we plug the $y_i$ terms in individually of the $c_i$ terms, that the expression will vanish identically because the $y_i$ terms are solutions to the homogeneous equation. The product rule on each term will be evaluated according to the Leibniz Rule, which is that $(c_i \cdot f_i)^{(n)}=\sum_{k=0}^n {n \choose k} c_i^{(k)} y_i(x)^{(n-k)}$. Now the $c_i y_i^{(n)}$ terms will vanish because we can factor out a $c_i$ and they will be exactly the homogeneous solution. Because the expression is identically equal to zero, the remaining terms must vanish as well. If we assume that each $\sum_{i=1}^n c_i' y_i^{(j)}=0$ for each j from 0 to n-2, then this will take care of this; the terms with higher derivatives on $c_i$ will also be 0, if this is true, then we do not need them for our derivation. In other words,
$c_1' y_1 + c_2' y_2 + \cdots + c_n' y_n = 0$
$c_n' y_1' + c_n' y_2' + \cdots + c_n' y_n' = 0$
$\vdots$
$c_n' y_1^{(n-2)} + c_n' y_2^{(n-2)} + \cdots + c_n' y_n^{(n-2)} = 0$.

So, turning back to our original ODE with the particular solution substituted in, we have
$(\sum_{i=1}^{n} c_i y_i)^{(n)} + a_{n-1}(\sum_{i=1}^{n} c_i y_i)^{(n-1)} + \dots + a_2(\sum_{i=1}^{n} c_i y_i)^{(2)} +$
$a_1(\sum_{i=1}^{n} c_i y_i)^{(1)} + a_0\sum_{i=1}^{n} c_i y_i = F(x)$.
But we know that most of the terms of this will vanish, from our assumption above. If we remove those terms, what remains is $\sum_{i=1}^{n} c_i' y_i^{(n-1)} = F(x)$. So this is where it is nice that I learned Cramer’s Rule literally days before learning how to do Variation of Parameters in the general case. We have a system of n equations (the n-1 from above, plus the one we just derived), of n unknowns (the $c_i$ terms). The determinant that we use here is used often enough to warrant a name: the Wronskian. We have that $c_i' = \frac{W_i(x)}{W(x)}$, or $c_i = \int \frac{W_i(x)}{W(x)}$, where $W_i(x)$ is the Wronskian of the fundamental system with the ith column replaced with $\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ F(x) \end{bmatrix}$. So we finally have $y_p = \sum_{i=1}^n \int \frac{W_i(x)}{W(x)} y_i$.

Well, that’s the theory, but as always here, that is only half of the story. A Wronskian function is already implemented in SymPy, and finding $W_i(x)$ simply amounts to $F(x)$ times the Wronskian of the system without the ith equation, all times $(-1)^i$. So implementing it was easy enough. But it soon became clear that there would be some problems with this method. Sometimes, the SymPy would return a really simple Wronskian, something like $-4e^{2x}$, but other times, it would return something crazy. For example, consider the expression that I reported in SymPy issue 1562. The expression is (thanks to SymPy’s latex() command, no thanks to WordPress’s stupid auto line breaks that have forced me to upload my own image. If it wasn’t such a pain, I would do it for every equation, because it looks much nicer.):
.
This is the Wronskian, as calculated by SymPy’s wronskian() function, of
$\begin{bmatrix}x \sin{x}, & \sin{x}, & 1, & x \cos{x}, & \cos{x}\end{bmatrix}$, which is the set of linearly independent solutions to the ODE ${\frac {d^{5}}{d{x}^{5}}}f \left( x \right) +2\,{\frac {d^{3}}{d{x}^{3}}}f \left( x \right) +{\frac {d}{dx}}f \left( x \right) -1$. Well, the problem here is that, as verified by Maple, that complex Wronskian above is identically equal to $-4$. SymPy’s simplify() and trigsimp() functions are not advanced enough to handle it. It turns out that in this case, the problem is that SymPy’s cancel() and factor() routines do not work unless the expression has only symbols in it, and that expression requires you to cancel and factor to find the $\cos^2{x} + \sin^2{x}$ (see the issue page for more information on this). Unfortunately, SymPy’s integrate() cannot handle that unsimplified expression in the denominator of something, as you could imagine, and it seems like almost every time that sin’s and cos’s are part of the solution to the homogeneous equation, the Wronskian becomes too difficult for SymPy to simplify. So, while I was hoping to slip along with only variation of parameters, which technically solves every linear inhomogeneous ODE, it looks like I am going to have to implement the method of undetermined coefficients. Variation of parameters will still be useful, as undetermined coefficients only works if the expression on the right hand side of the equation, $F(x)$ has a finite set of linearly independent derivatives (such as sin, cos, exp, polynomial terms, and combinations of them (I’ll talk more about this whenever I implement it).

The good news here is that I discovered that I was wrong. I had previously believed that among the second order special cases were cases that could only be handled by variation of parameters or undetermined coefficients, but it turns out I was wrong. All that was implemented were the homogeneous cases for second order linear with constant coefficients. In addition to this, there was one very special case ODE that Ondrej had implemented for an example (examples/advanced/relativity.py). The ODE is
$-2({\frac{d}{dx}}f(x)){e^{-f(x)}}+x({\frac{d}{dx}}f(x))^{2}{e^{-f(x)}}-x({\frac{d^{2}}{d{x}^{2}}}f(x)){e^{-f(x)}}$, which is the second derivative of $xe^{-f(x)}$ with respect to x. According to the example file, it is know as Einstein’s equations. Maple has a nice odeadvisor() function similar to the classify_ode() function I am writing for SymPy that tells you all of the different ways that an ODE can be solved. So, I plugged that ODE into it and got a few possible methods out that I could potentially implement in SymPy to maintain compatibility with the example equation. The chief one is that the lowest order of f in the ODE is 1 (assuming you divide out the $e^{-f(x)}$ term, which is perfectly reasonable as that term will never be 0. You can then make the substitution $u = f'(x)$, and you will reduce the order of the ODE to first order, which in this case would be a Bernoulli equation, the first thing that I ever implemented in SymPy.

But I didn’t do that. Reduction of order methods would be great to have for dsolve(), but that is a project for another summer. Aside from that method, Maple’s odeadvisor() also told me that it was a Liouville ODE. I had never heard of that method, and neither it seems has Wikipedia or “Uncle Google” (as Ondrej calls it). Fortunately, Maple’s Documentation has a nice page for each type of ODE returned by odeadvisor(), so I was able to learn the method. The method relies on Lie Symmetries and exact second order equations, neither of which I am actually familiar with, so I will not attempt to prove anything here. Suffice it to say that if an ODE has the form
${\frac{d^{2}}{d{x}^{2}}}y(x)+g(y(x))({\frac{d}{dx}}y(x))^{2}+f(x){\frac{d}{dx}}y(x)=0$, then the solution to the ODE is
$\int^{y(x)}{e^{\int g(a){da}}}{da}+C1\int{e^{-\int f(x){dx}}}{dx}+C2=0$
You could probably verify this by substituting the solution into the original ODE. See the Maple Documentation page on Liouville ODEs, as well as the paper they reference (Goldstein and Braun, “Advanced Methods for the Solution of Differential Equations”, see pg. 98).

The solution is very straight forward–as much so as first order linear or Bernoulli equations, so it was a cinch to implement it. It looks like quite a few differential equations generated by doing $F''(y(x), x)$ for some function or x and y $F(y(x), x)$ generates equations of that type, so it could be actually useful for solving other things.

Before I sign off, I just want to mention one other thing that I implemented. I wanted my linear homogeneous constant coefficient ODE solver to be able to handle ODEs for which SymPy can’t solve the characteristic equation, for whatever reason. SymPy has RootOf() objects similar to Maple that let you represent the roots of a polynomial without actually solving it, or even being able to solve it, but a you can only use RootOf’s if you know that none of the roots are repeated. Otherwise, you would have to know which terms require an additional $x^i$ to preserve linear independence. Well, it turns out that there is a way to tell if a polynomial has repeated roots without solving for them. There is a number associated with every polynomial of one variable called the discriminant. For example, the discriminant of the common quadratic polynomial $ax^2 + bx + c$ is the term under the square root of the famous solution $b^2 - 4ac$. It is clear that a quadratic has repeated roots if and only if the discriminant is 0. Well, the same is true for the discriminant of any polynomial. I am not highly familiar with this (ask me again after I have taken my abstract algebra class next semester), but apparently there is something called the resultant, which is the product of the differences of roots between two polynomials and which can also be calculated without explicitly finding the roots of the polynomials. Clearly, this will be 0 if and only if the two polynomials share a root. So the discriminant is built from the fact that a polynomial has a repeated root iff it shares a root with its resultant. So it is basically the resultant of a polynomial and its derativave, times an extra factor. It is 0 if and only if the polynomial has a repeated root.

Fortunately, SymPy’s excelent Polys module already had resultants implemented (quite efficiently too, I might add), so it was easy to implement the discriminant. I added it as issue 1555. If you are a SymPy developer and you have somehow managed to make yourself read this far (bless your heart), please review that patch.

Well, this has turned out to be one hella long blog post. But what can I say. You don’t have to read this thing (except for possibly my mentor. Sorry Andy). And I haven’t been quite updating weekly like I am supposed to be, so this compensates. If you happened upon this blog post because, like me, you were looking for a general treatment of variation of parameters, I hope you found my little write up helpful. And if you did, and you now understand it, could you go ahead and improve the Wikipedia article. I’m not up to it?