Undetermined Coefficients

August 17, 2009

[Sorry for the delay in this post. I was having some difficulties coming up with some of the rationales below. Also, classes have started, which has made me very busy.]

If there was one ODE solving method that I did not want to implement this summer, it was undetermined coefficients. I didn’t really like the method too much when we did it my my ODE class (though it was not as unenjoyable as series methods). The thing that I never really understood very well is to what extent you have to multiply terms in the trial function by powers of x to make them linearly independent of the solution to the general equation. We did our ODEs homework in Maple, so I would usually just keep trying higher powers of x until I got a solution. But to implement it in SymPy, I had to have a much better understanding of the exact rules for it.

From a user’s point of view, the method of undetermined coefficients is much better than the method of variation of parameters. While it is true that variation of parameters is a general method and undetermined coefficients only works on a special class of functions, undetermined coefficients requires no integration or advanced simplification, so it is fast (very fast, as well shall see below). All that the CAS has to do is figure out what a trial function looks like, plug it into the ODE, and solve for the coefficients, which is a system of linear equations.

On the other hand, from the programmer’s point of view, variation of parameters is much better. All you have to do is take the Wronskian of the general solution set and use it to set up some integrals. But the Wronskian has to be simplified, and if the general solution contains sin’s and cos’s, this requires trigonometric simplification not currently available in SymPy (although it looks like the new Polys module will be making a big leap forward in this area). Also, integration is slow, and in SymPy, it often fails (hangs forever).

Figuring out what the trial function should be for undetermined coefficients is way more difficult to program, but having finnally finished it, I can say that it is definitely worth having in the module. Problems that it can solve can run orders of magnitude faster than the variation of parameters, and often variation of parameters can’t do the integral or returns a less simplified result.

So what is this undetermined coefficients? Well, the idea is this: if you knew what each linearly independent term of the particular solution was, minus the coefficients, then you could just set each coefficient as an unknown, plug it into the ODE, and solve for them. It turns out that resulting system of equations is linear, so if you do the first part right, you can always get a solution.

The key thing here is that you know what form the particular solution will take. However, you don’t really know this ahead of time. All you have is the linear ode $a_ny^{(n)}(x) + \dots + a_1y'(x) + a_0y(x) = F(x)$ (as far as I can tell, this only works in the case where the coefficients $a_i$ are constant with respect to x. I’d be interested to learn that it works for other linear ODEs. At any rate, that is the only one that works in my branch right now.). The solution to the ode is $y(x) = y_g(x) + y_p(x)$, where $y_g(x)$ is the solution to the homogeneous equation $f(x) \equiv 0$, and $y_p(x)$ is the particular solution that produces the $F(x)$ term on the right hand side. The key here is just that. If you plug $y_p(x)$ into the left hand side of the ode, you get $F(x)$.

It turns out that this method only works if the function $F(x)$ only has a finite number of linearly independent derivatives (I am unsure, but this might be able to work in other cases, but it would involve much more advanced mathematics). So what kind of functions have a finite number of linearly independent solutions? Obviously, polynomials do. So does $e^x$, $\cos{x}$, and $\sin{x}$. Also, if we multiply two or more of these types together, then we will get a finite number of linearly independent solutions after applying the product rule. But is that all? Well, if we take the definition of linear independence from linear algebra, we know that a set of n vectors $\{\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}, \dots, \boldsymbol{v_n}\}$, not all zero, are linearly independent only if $a_1\boldsymbol{v_1} + a_2\boldsymbol{v_2} + a_3\boldsymbol{v_3} + \dots + a_n\boldsymbol{v_n}=0$ holds only when $a_1 \equiv 0, a_2 \equiv 0, a_3 \equiv 0, \dots, a_n \equiv 0$, that is, the only solution is the trivial one (remember, this is the definition of linear independence). They are linearly dependent if there exist weights $a_1, a_2, a_3, \dots, a_n$, not all 0, such that the equation $a_1\boldsymbol{v_1} + a_2\boldsymbol{v_2} + a_3\boldsymbol{v_3} + \dots + a_n\boldsymbol{v_n}=0$ is satisfied. Using this definition, we can see that a function $f(x)$ will have a finite number of linearly independent derivatives if it satisfies $a_nf^{(n)}(x) + a_{n - 1}f^{(n - 1)}(x) + \dots + a_1f'(x) + a_0f(x) = 0$ for some $n$ and with $a_i\neq 0$ for some $i$. But this is just a homogeneous linear ODE with constant coefficients, which we know how to solve. The solutions are all of the form $ax^ne^{b x}\cos{cx}$ or $ax^ne^{b x}\sin{cx}$, where a, b, and c are real numbers and n is a non-negative integer. We can set the various constants to 0 to get the type we want. For example, for a polynomial term, b will be 0 and c will be 0 (use the cos term).

So this gives us the exact form of functions that we need to look for to apply undetermined coefficients, based on the assumption that it only works on functions with a finite number of linearly independent derivatives.

Well, implementing it was quite difficult. For every ODE, the first step in implementation is matching the ODE, so the solver can know what methods it can apply to a given ODE. To match in this case, I had to write a function that determined if the function matched the form given above, which was not too difficult, though not as trivial as just grabbing the right hand side in variation of parameters. The next step is to use the matching to format the ODE for the solver. In this case, it means finding all of the finite linearly independent derivatives of the ODE, so that the solver can just create a linear combination of them solve for the coefficients. This was a little more difficult, and it took some lateral thinking.

At this point, there is one more thing that needs to be noted. Since the trial functions, that is, the linearly independent derivative terms of the right hand side of the ODE, are of the same form as the solutions to the homogeneous equation, it is possible that one of the trial function terms will be a solution to the homogeneous equation. If this happens, plugging it into the ODE will cause it to go to zero, which means that we will not be able to solve for a coefficient for that term. Indeed, that term will be of the form $C1*\textrm{term}$ in the final solution, so even if we had a coefficient for it, it would be absorbed into this term from the solution to the homogeneous equation. For example, variation of parameters will give a coefficient for such terms, even though it is unnecessary. This is a clue that Maple uses variation of parameters for all linear constant coefficient ODE solving, because it gives the unnecessary terms with the coefficients that would be given by variation of parameters, instead of absorbing them into the arbitrary constants.

We can safely ignore these terms for undetermined coefficients, because their coefficients will not even appear in the system of linear equations of the coefficients anyway. But, without these coefficients, we will run into trouble. It turns out that if a term $x^ne^{ax}\sin{bx}$ or $x^ne^{ax}\cos{bx}$ is repeated solution to the homogeneous equation, and $x^{n + 1}e^{ax}\sin{bx}$ or $x^{n + 1}e^{ax}\cos{bx}$ is not, so that $n$ is the highest $x$ power that makes it a solution to the homogeneous equation, and if the trial solution has $x^me^{ax}\sin{bx}$ or $x^me^{ax}\cos{bx}$ terms, but not $x^{m + 1}e^{ax}\sin{bx}$ or $x^{m + 1}e^{ax}\cos{bx}$ terms, so that $m$ is the highest power of $x$ in the the trial function terms, then we need to multiply these trial function terms by $x^{n + m}$ to make them linearly independent with the solutions of the homogeneous equation.

Most references simply say that you need to multiply the trial function terms by “sufficient powers of x” to make them linearly independent with the homogeneous solution. Well, this is just fine if you are doing it by hand or you are creating the trial function manually in Maple and plugging it in and solving for the coefficients. You can just keep upping the powers of x until you get a solution for the coefficients. Creating those trial functions in Maple, plugging them into the ODE, and solving for the coefficients is exactly what I had to do for my homework when I took ODEs last spring, and this “upping powers” trial and error method is exactly the method I used. But when you are doing it in SymPy, you need to know exactly what power to multiply it by. If it is too low, you will not get solution to the coefficients. If it is too high, you can actually end up with too many terms in the final solution, giving a wrong answer.

Fortunately, my excellent ODEs textbook gives the exact cases to follow, and so I was able to implement it correctly. The textbook also gives a whole slew of exercises, all for which the solutions are given. As usual, this helped me to find the bugs in my very complex and difficult to write routine. It also helped me to find a match bug that would have prevented dsolve() from being able to match certain types of ODEs. The bug turned out to be fundamental to the way match() is written, so I had to write my own custom matching function for linear ODEs.

The final step in solving the undetermined coefficients is of course just creating a linear combination of the trial function terms, plugging it into the original ODE, and setting the coefficients of each term on each side equal to each other, which gives a linear system. SymPy can solve these easily, and once you have the values of the coefficients, you can use them to build your particular solution, at which point, you are done.

The results were astounding. Variation of parameters would hang on many simple inhomogeneous ODEs because of poor trig simplification of the Wronsikan, but my undetermined coefficients method handles them perfectly. Also, there is no need to worry about absorbing superfluous terms into the arbitrary constants as with variation of parameters, because they are removed from within the undetermined coefficients algorithm.

But the biggest thing was speed. Here are some benchmarks on some random ODEs from the test suite. WordPress code blocks are impervious to whitespace, as I have mentioned before, so no pretty printing here. Also, it truncates the hints. The hints used are 'nth_linear_constant_coeff_undetermined_coefficients' and 'nth_linear_constant_coeff_variation_of_parameters':

 In [1]: time dsolve(f(x).diff(x, 2) - 3*f(x).diff(x) - 2*exp(2*x)*sin(x), f(x), hint='nth_linear_constant_coeff_undetermined_coefficients') CPU times: user 0.07 s, sys: 0.00 s, total: 0.08 s Wall time: 0.08 s Out[2]: f(x) == C1 + (-3*sin(x)/5 - cos(x)/5)*exp(2*x) + C2*exp(3*x) In [3]: time dsolve(f(x).diff(x, 2) - 3*f(x).diff(x) - 2*exp(2*x)*sin(x), f(x), hint='nth_linear_constant_coeff_variation_of_parameters') CPU times: user 0.92 s, sys: 0.01 s, total: 0.93 s Wall time: 0.94 s Out[4]: f(x) == C1 + (-3*sin(x)/5 - cos(x)/5)*exp(2*x) + C2*exp(3*x) In [5]: time dsolve(f(x).diff(x, 4) - 2*f(x).diff(x, 2) + f(x) - x + sin(x), f(x), hint='nth_linear_constant_coeff_undetermined_coefficients') CPU times: user 0.06 s, sys: 0.00 s, total: 0.06 s Wall time: 0.06 s Out[6]: f(x) == x - sin(x)/4 + (C1 + C2*x)*exp(x) + (C3 + C4*x)*exp(-x) In [7]: time dsolve(f(x).diff(x, 4) - 2*f(x).diff(x, 2) + f(x) - x + sin(x), f(x), hint='nth_linear_constant_coeff_variation_of_parameters') CPU times: user 5.43 s, sys: 0.03 s, total: 5.46 s Wall time: 5.52 s Out[8]: f(x) == x - sin(x)/4 + (C1 + C2*x)*exp(x) + (C3 + C4*x)*exp(-x) In [9]: time dsolve(f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - sin(x) - cos(x), f(x), 'nth_linear_constant_coeff_undetermined_coefficients') CPU times: user 0.10 s, sys: 0.00 s, total: 0.10 s Wall time: 0.11 s Out[10]: f(x) == C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2 In [11]: time dsolve(f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - sin(x) - cos(x), f(x), 'nth_linear_constant_coeff_variation_of_parameters') 

The last one involves a particularly difficult Wronskian for SymPy (run it with hint=’nth_linear_constant_coeff_variation_of_parameters_Integral’, simplify=False).

Wall time comparisons reveal amazing speed differences. We’re talking orders of magnitude.

 In [13]: 0.94/0.08 Out[13]: 11.75 In [14]: 5.52/0.06 Out[14]: 92.0 In [15]: oo/0.11 Out[15]: +inf 

Of course, variation of parameters has the most difficult time when there are sin and cos terms involved, because of the poor trig simplification in SymPy. So let’s see what happens with an ODE that just has exponentials and polynomial terms involved.

 In [16]: time dsolve(f(x).diff(x, 2) + f(x).diff(x) - x**2 - 2*x, f(x), hint='nth_linear_constant_coeff_undetermined_coefficients') CPU times: user 0.10 s, sys: 0.00 s, total: 0.10 s Wall time: 0.10 s Out[17]: f(x) == C1 + x**3/3 + C2*exp(-x) In [18]: time dsolve(f(x).diff(x, 2) + f(x).diff(x) - x**2 - 2*x, f(x), hint='nth_linear_constant_coeff_variation_of_parameters') CPU times: user 0.19 s, sys: 0.00 s, total: 0.19 s Wall time: 0.20 s Out[19]: f(x) == C1 + x**3/3 + C2*exp(-x) In [20]: time dsolve(f(x).diff(x, 3) + 3*f(x).diff(x, 2) + 3*f(x).diff(x) + f(x) - 2*exp(-x) + x**2*exp(-x), f(x), hint='nth_linear_constant_coeff_undetermined_coefficients') CPU times: user 0.09 s, sys: 0.00 s, total: 0.09 s Wall time: 0.09 s Out[21]: f(x) == (C1 + C2*x + C3*x**2 + x**3/3 - x**5/60)*exp(-x) In [22]: time dsolve(f(x).diff(x, 3) + 3*f(x).diff(x, 2) + 3*f(x).diff(x) + f(x) - 2*exp(-x) + x**2*exp(-x), f(x), hint='nth_linear_constant_coeff_variation_of_parameters') CPU times: user 0.29 s, sys: 0.00 s, total: 0.29 s Wall time: 0.29 s Out[23]: f(x) == (C1 + C2*x + C3*x**2 + x**3/3 - x**5/60)*exp(-x) 

The wall time comparisons here are:

 In [24]: 0.20/0.10 Out[24]: 2.0 In [25]: 0.29/0.09 Out[25]: 3.22222222222 

So we don’t have orders of magnitude anymore, but it is still 2 to 3 times faster. Of course, most ODEs of this form will have sin or cos terms in them, so the order of magnitude improvement over variation of parameters can probably be attributed to undetermined coefficients in general.

Of course, we know that variation of parameters will still be useful, because functions like $\ln{x}$, $\sec{x}$ and $\frac{1}{x}$ do not have a finite number of linearly independent derivatives, and so you cannot apply the method of undetermined coefficients to them.

There is one last thing I want to mention. You can indeed multiply any polynomial, exponential, sin, or cos functions together and still get a function that has a finite number of linearly independent solutions, but if you multiply two or more of the trig functions, you have to apply the power reduction rules to the resulting function to get it in terms of sin and cos alone. Unfortunately, SymPy does not yet have a function that can do this, so to solve such a differential equation with undetermined coefficients (recommended, see above), you will have to apply them manually yourself. Also, just for the record, it doesn’t play well with exponentials in the form of sin’s and cos’s or the other way around (complex coefficients on the arguments), so you should back convert those first too.

Well, this concludes the first of two blog posts that I promised. I also promised that I would write about my summer of code experiences. Not only is this important to me, but it is a requirement. I really hope to get this done soon, but with classes, who knows.

Los Alamos “Sprint”

August 17, 2009

Last weekend, Luke came to visit Ondrej in Los Alamos, so I decided to drive him up from Albuquerque and visit him again. It was nice meeting Luke and seeing Ondrej again.

Aside from coding (the main thing that I did was fix an ugly match bug that was preventing dsolve() from recognizing certain ODEs), we visited the atomic museum in Los Alamos, the Valles Caldera, and some of the surrounding hot springs.

Here are some pictures that Luke took with his iPhone. Stupid WordPress seems to insist on flipping some of them (I can’t fix it):

This is one of three posts that I plan on doing this week. I just finished my GSoC project today/last night, so I will be blogging about that. I plan on doing a post on the method of Undetermined Coefficients, as well as some other things that I managed to do. The other post will be my general musings/advice for GSoC. That will probably be my last post here in a while. I plan on continuing work with SymPy, but I get very busy with classes, so I most likely won’t be doing much until next summer.

Testing implicit solutions to ODEs

August 12, 2009

So, the hard deadline for GSoC it Monday, so this will probably be my last post until then (I am very busy trying to finish up the ode module by then). But this is one of those things that you just have to blog about.

So I have this checksol function in test_ode.py that attempts to check it the solutions to odes are valid or not. It was a relic of the old ode module. For that, it would just substitute the solution into the ode and see if it simplified to 0. That is what it still does, if the solution is solved for f(x) (the function for all of my ode tests). But if the solution is implicit in f, either because solve() is not good enough to solve it or because it cannot be solved, then that method obviously does not work. So what I was trying to do is what my textbook suggested. Take the derivative of the solution implicitly n times, where n is the order of the ode, and see if that is equal to the ode. Basically, I was subtracting the ode from it and seeing if it reduced to 0.

However, it wasn’t really working at all for most of my implicit solutions, even the really simple ones. I ended up XFAILing most of my implicit checksol tests. I think every single homogeneous coefficients had an implicit solution, and none of them were working with checksol().

So I started to ask around on IRC to see if anyone had any better ideas for testing these. Ondrej couldn’t think of anything. Luke and Chris worked on an example that I gave them, and it seemed to be that it wasn’t correct (which I didn’t believe for a second, because the solution was straight out of my text, and both homogeneous coefficients integrals produced that same solution). It turns out that we were mixing up $\log{\frac{y}{x}}$ and $\log{\frac{x}{y}}$ terms. One of those appeared in the ode and the other appeared in the solution (the ode was $y dx + x\log{\frac{y}{x}}dy - 2x dy = 0$ and the solution is $\frac{y}{1 + \log{\frac{x}{y}}}=C$, number 9 from my odes text, pg. 61.

So Chris had a novel idea. For 1st order odes, you can take the derivative of the solution and solve for $\frac{dy}{dx}$, which will always be possible, because differentiation is a linear operator. Then substitute that into the original ode, and it will reduce.

So we were talking about this on IRC later, and I had an epiphany as to why my original method wasn’t working. After trying it manually on an ode, I found that I had to multiply through the solution’s derivative by $\frac{x}{f(x)}$ to make it equal to the ode. Then, that reminded me of an important solution method that I didn’t have time to implement this summer: integrating factors. I remember that my textbook had mentioned that there is a theorem that states that every 1st order ODE that is linear in the derivative has a unique integrating factor that makes it exact. And I realized, the derivative of the solution will be equal to the ODE if and only if the ODE is exact. I checked my exact tests and verified my hunch. I had to XFAIL all of my implicit homogeneous coefficients solutions, but all of my exact checksols were working just fine.

So I refactored my checksol function to do this, and it now can check almost every one of my failing checksols. The exceptions are some where trigsimp() cannot simplify the solution to 0 (we have a poor trigsimp), a second order solution (the above trick only works on 1st order odes, I believe), and some other simplification problems.

The only down side to this new routine is that it is kind of slow (because of the simplification). I am going to have to skip a test of only 6 solutions because it takes 24 seconds to complete.

Homogeneous coefficients corner case

August 10, 2009

Before I started the program, I implemented Bernoulli equations. But the general solution to Bernoulli equations involves raising something to the power of $\frac{1}{1-n}$, where n is the power of the dependent term (see the Wikipedia page for more info). This works great, as I soon discovered, unless n == 1. Then you get something to the power of $\infty$. So I had to go in and remove the corner case.

So you think that after that I would have been more careful after that about checking that if general solution that divides by something I would test to see if that something is not zero before returning it as a solution.

Well, as I was just trying to implement some separable equation tests, I was going through the exercises of my ode text as I usually do for tests, and I came across $xy' - y = 0$. If you recall, this equation also has coefficients that homogeneous of the same order (1). From the general solution to homogeneous coefficients, you would plug it into $\int{\frac{dx}{x}}=\int{\frac{-Q(1,u)du}{P(1,u)+uQ(1,u)}}+C$ where $u = \frac{y}{x}$ or $\int{\frac{dy}{y}}=\int{\frac{-P(u,1)du}{uP(u,1)+Q(u,1)}}+C$ where $u = \frac{x}{y}$ (here, P and Q are from the general form $P(x,y)dx+Q(x,y)dy=0$). Well, it turns out that if you plug the coefficients from my example into those equations, the denominator will become 0 for each one. So I (obviously) need to check for that $P(1,u)+uQ(1,u)$ and $uP(u,1)+Q(u,1)$ are not 0 before running the homogeneous coefficients solver on a differential equation.

Variation of Parameters and More

August 1, 2009

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?