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> And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc). >

> And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc). > Because in this way you can obtain solution to any DE with any degree of precision (within reasonable assumptions about complexity of the problem). This is actually an entirely different class of thing, and it's important not to confuse them. That's kind of like saying, "why would I ever need a function that can compute square roots at run-time? Just tell me whatever number it is you want the square root of, and then using my pocket calculator, we can hard-code its square root to any number of digits of accuracy at compile-time!" A numerical solution is a specific answer only to a specific input; depending on what you're trying to do, that might be useful or it might be absolutely useless. An analytical solution is like a short "run-time" program that computes solutions for given parameters, and which can often be reversed and inverted. One example where it's especially painful is when you're trying to do the inverse: find the correct differential equation that results in a desired solution. (AKA a design problem: you need to design a filter, oscillator, linkage, control law, etc, that results in a desired behaviour). This isn't just about ivory tower mathematicians! Engineers need to do this! An analytic solution makes this easy. Imagine if you, an electrical engineer, only know how to be handed a given schematic, break it down into the DE's that govern it, and then numerically solve them. You wouldn't be able to design a new circuit, just predict the behaviour of existing ones. Imagine needing to run sweeps of SPICE simulations over wide ranges of resistor and capacitor values to design a low-pass filter just because you never learned how an RC filter works. It's not that you can't do that, and in some cases where you're dealing with nonlinear systems it's the most pragmatic approach. But that's also a very simple example; as the system gets more complex, the curse of dimensionality will make that more and more challenging, while a system of linear differential equations could continue to have exact solutions no matter how many dimensions you have, and furthermore, make it easy to pick out exactly which variables are the most critical. Or maybe you want to find an optimum point. You can do numeric optimization, but again, you'll get hit with the same curse of dimensionality if the system gets complicated. But if you have an algebraic solution, you might be able to pick out a formula for it. Then not only will you get the optimum (either its coordinates, or its value), you also get an equation that lets you see, in functional form, how that optimum point moves around as you change the other variables in the problem. Then maybe you can maximize that if you want to, and so on. In other cases, being able to remove even one dimension from the analysis can be used to speed up a numerical solution over all the other parameters. Like if you're designing a plane and you have to search over various wingspans, pitches, taper, and so on... if you can get an analytic equation that gives you an optimum wingspan for a given pitch and taper, then you can reduce your search space by an entire dimension, which has a huge impact.

BeetleB 6y ago
While I would generally agree with the sentiment, I disagree with you on diff eq. The standard education in a diff eq course is almost useless for real world problems (beyond the obvious linear diff equations). Much of what is taught in such a course is never used - even by mathematicians - including applied (Google rants on integrating factors). Real world diff equations are notoriously difficult to deal with analytically, and usually a numerical approach is the only tool in the toolbox.
- jbay808 6y ago
  Real world nonlinear differential equations are notoriously difficult. Linear ODEs and PDEs are not, even in many dimensions, and many important real-world systems (building blocks of electronic circuits, acoustics and vibrations, optics, and so on) are linear, or are easily approximated as such. Yes, for complicated nonlinear high-order systems, you need a numeric solution and when that's the case, that's fine. But for engineers, it's really important to identify when that's not the case. This is really fundamental. The whole reason we can even define things like "impedances", "natural frequencies", "buckling modes", "reflection coefficients", "critical speeds", and "damping ratios" that let you actually design things like filters, speakers, ultrasound systems, lenses, motors, transmission lines, delay lines, antennas, rockets, and the Eiffel Tower, is because you can solve the governing equation analytically and then collect the solution and recognize an important term that factors out, get the formula for it, give it a name, and take note of the variables that it depends on. Even for something as infamous as the Navier-Stokes equation, you can extract important parameters like the Reynolds number, Froude number, and so on -- which I hope I don't need to explain why they're extremely useful, even for a numerical design approach -- so even if you can't solve the equation, you can still get a lot of mileage out of the analysis itself. (Even integrating factors are something that I've used "in anger" in my career, as part of an assignment to predict inventory stockouts).
  - BeetleB 6y ago
    Nonlinear differential equations are more common in the real world than linear ones. Our textbooks tend to have a lot more linear ones simply because they are easier to handle and teach, not because of their prevalence. Even when you get to the nitty gritty of real world circuits, you'll encounter nonlinearities and have to deal with them. The linear approach is fine for undergrad curricula, but having worked with some of the frontiers of circuit/semiconductor work professionally - you can't build great products by relying on linear diff eqs alone. Furthermore, linear diff eq's are somewhat of a solved problem. A researcher is unlikely to discover a phenomenon today that is governed by a linear equation. > Even for something as infamous as the Navier-Stokes equation, you can extract important parameters like the Reynolds number, Froude number, and so on -- which I hope I don't need to explain why they're extremely useful, even for a numerical design approach -- so even if you can't solve the equation, you can still get a lot of mileage out of the analysis itself. Here I think we are merging a little: I took only one fluid mechanics/dynamics course (albeit 20 years ago), and we were taught the Reynolds number, etc. The thing is, I would say there were only two engineering courses I took in my whole undergrad where the analytical approach is not particularly helpful, and one really has to go full out "engineer" - either by looking up lots of (empirical) graphs and tables, or use numerical methods. Fluid mechanics was one of those courses. Yes, I agree that doing some level of analytical work on the diff equations is very useful, but in the majority of cases, the main purpose of doing some theoretical analysis on the differential equation is to get it into a form where one can apply numerical techniques to it. I'm not a fluid mechanics person, but I'd wager that most of the useful results we have from the Navier Stokes equation came out of some numerical technique (perhaps after a bit of analytical work). In any case: Yes, I agree - it is useful to do some analytical work on it. But the bulk of the gains in the real world come from applying numerical technique on them. And more to the point: Most of what is taught in a typical diff eq course is of little use in doing that analytical work. Edit: As another anecdote, for my grad research, I went the analytical route, as that was what my advisor preferred. Thing is, it was fine in his day when the field was young. Most of the people doing research use numerical methods (often from first principles, so almost no analytical approach applied). They basically "won". When you look at the useful results in the discipline in the last 10-15 years, the numerical ones outnumber the analytical ones easily by a factor of 10.

BeetleB 6y ago

While I would generally agree with the sentiment, I disagree with you on diff eq. The standard education in a diff eq course is almost useless for real world problems (beyond the obvious linear diff equations). Much of what is taught in such a course is never used - even by mathematicians - including applied (Google rants on integrating factors). Real world diff equations are notoriously difficult to deal with analytically, and usually a numerical approach is the only tool in the toolbox.

jbay808 6y ago
Real world nonlinear differential equations are notoriously difficult. Linear ODEs and PDEs are not, even in many dimensions, and many important real-world systems (building blocks of electronic circuits, acoustics and vibrations, optics, and so on) are linear, or are easily approximated as such. Yes, for complicated nonlinear high-order systems, you need a numeric solution and when that's the case, that's fine. But for engineers, it's really important to identify when that's not the case. This is really fundamental. The whole reason we can even define things like "impedances", "natural frequencies", "buckling modes", "reflection coefficients", "critical speeds", and "damping ratios" that let you actually design things like filters, speakers, ultrasound systems, lenses, motors, transmission lines, delay lines, antennas, rockets, and the Eiffel Tower, is because you can solve the governing equation analytically and then collect the solution and recognize an important term that factors out, get the formula for it, give it a name, and take note of the variables that it depends on. Even for something as infamous as the Navier-Stokes equation, you can extract important parameters like the Reynolds number, Froude number, and so on -- which I hope I don't need to explain why they're extremely useful, even for a numerical design approach -- so even if you can't solve the equation, you can still get a lot of mileage out of the analysis itself. (Even integrating factors are something that I've used "in anger" in my career, as part of an assignment to predict inventory stockouts).
- BeetleB 6y ago
  Nonlinear differential equations are more common in the real world than linear ones. Our textbooks tend to have a lot more linear ones simply because they are easier to handle and teach, not because of their prevalence. Even when you get to the nitty gritty of real world circuits, you'll encounter nonlinearities and have to deal with them. The linear approach is fine for undergrad curricula, but having worked with some of the frontiers of circuit/semiconductor work professionally - you can't build great products by relying on linear diff eqs alone. Furthermore, linear diff eq's are somewhat of a solved problem. A researcher is unlikely to discover a phenomenon today that is governed by a linear equation. > Even for something as infamous as the Navier-Stokes equation, you can extract important parameters like the Reynolds number, Froude number, and so on -- which I hope I don't need to explain why they're extremely useful, even for a numerical design approach -- so even if you can't solve the equation, you can still get a lot of mileage out of the analysis itself. Here I think we are merging a little: I took only one fluid mechanics/dynamics course (albeit 20 years ago), and we were taught the Reynolds number, etc. The thing is, I would say there were only two engineering courses I took in my whole undergrad where the analytical approach is not particularly helpful, and one really has to go full out "engineer" - either by looking up lots of (empirical) graphs and tables, or use numerical methods. Fluid mechanics was one of those courses. Yes, I agree that doing some level of analytical work on the diff equations is very useful, but in the majority of cases, the main purpose of doing some theoretical analysis on the differential equation is to get it into a form where one can apply numerical techniques to it. I'm not a fluid mechanics person, but I'd wager that most of the useful results we have from the Navier Stokes equation came out of some numerical technique (perhaps after a bit of analytical work). In any case: Yes, I agree - it is useful to do some analytical work on it. But the bulk of the gains in the real world come from applying numerical technique on them. And more to the point: Most of what is taught in a typical diff eq course is of little use in doing that analytical work. Edit: As another anecdote, for my grad research, I went the analytical route, as that was what my advisor preferred. Thing is, it was fine in his day when the field was young. Most of the people doing research use numerical methods (often from first principles, so almost no analytical approach applied). They basically "won". When you look at the useful results in the discipline in the last 10-15 years, the numerical ones outnumber the analytical ones easily by a factor of 10.