Differential equations - definitions

I am having a great deal of trouble with the definitions used throughout the book so far - i.e. linear, homogenous, non-homogenous, etc. I am not sure why exactly they are useful to know. I am having trouble seeing any reason to label one as a particular form, if all that is needed is to see the form and know what procedure might be used to solve it. Other than perhaps years from now, looking up "how to solve x-type definition problem" by the definition?

For instance, in calculus, if I see a problem that must be solved with a particular procedure, it is because I have recognized a form. So I go look up the procedure (ex: integration by parts), or the Heaviside cover up procedure. Most of the techniques are named after procedures, if I recall, like there wasn't a definition for trigonometric substitution forms, you just knew the form and the procedure for it, with no additional label layer in between needed. I don't recall such a huge emphasis on definitional labeling in mathematics before, but also I am still an undergraduate! :)

Does anyone have any recommendations on how to understand these definitions better? Most everything online and in my books keeps referring to long symbol salad strings and I am a bad memorizer. I remember the motions of the procedures and the shapes they take and know why I do such a thing such a way, and I am scared this is going to be an achilles heel for me in a huge way.

If it is relevant for a response, I am a returning student working towards a BS in Physics and I am in my differentials/linear equations/physics 2 part of my educational path.

• The best way to remember definitions (for many) is to find out what they "really mean", i.e., find out where they come from, see lots of examples of objects they apply to, and see how they are useful. For this you can either a) push onwards with the course, and all will (hopefully) be revealed or b) perhaps try looking on math.stackexchange. If you don't find much helpful, you might try asking a (suitably phrased) question over there about what the motivations are behind these definitions and asking what other sources (like good books or lecture notes) you might consult for this information. – Will R Jan 28 '17 at 21:30
• Good advice, I will work on how to phrase it so I can be less frustrated sounding than I currently am, as well. I originally didn't know which forum to start in with my question, so I tried here first and I am glad for that. I am going to compile a "examples" dictionary for my definitions as I go too, based on what you have said here. – The Schroedinger Cat Jan 29 '17 at 3:08
• Yes! "Example dictionary" is a very good part of working toward this understanding! And be sure to note the salient_properties of the examples that caused someone to name them in the way that they are named. – paul garrett Jan 29 '17 at 15:57
• I think learning to solve the equations is more important than to categorize them (in a normal ODE course). Yes, we do have some labels for the types of equations but big deal. And it helps in knowing how to solve them. second order, constant coefficient homo and nonhomo is the most important/common one for applications. Knowing how to recognize it is helpful. The homo/non-homo helps with constructing solutions (as you have to know how to do homo to do nonhomo--it is a part of the solution). But this is a nice feature as you are dividing a problem into parts. – guest Apr 7 '18 at 23:33

One reason that calculus solution concepts didn't necessarily have labels of this kind is that such things are fairly ad-hoc. But all of the terms you listed - "linear", "homogeneous", etc. - have deep meaning in the algebra of solutions. (Indeed, given your current state of studies, you should be asking about the relationship between differential equations and linear algebra, which might clarify this for you).

As an example, a homogeneous equation has the very nice property that if $f$ is a solution, so is any constant multiple of $f$. That seems convenient, I hope! (Not to mention the connection to linear algebra, but again I'm not sure if you have encountered that yet.) So it's not just the way in which you solve it, but the relationship the solutions have to each other. As this example shows, both types might have an infinite number of solutions with just one constant, but clearly the second one can't be manipulated in the same way as the first.

And it's going to be the same with most other terminology now that you are past calculus. But it's not bad, it's actually good; structure is awesome and useful! Good luck.

• Thank you kindly. Unfortunately, I mistakenly took Differentials by themselves, not knowing enough to know to take them together, and neither did my advisor! So, I am going to dig through some linear algebra, which I will need to teach myself anyway, and see if I can shed some light on things. Also picked up Ordinary differential equations by Tenenbaum, and it defines things a little differently than my class book, which helps. :) – The Schroedinger Cat Jan 29 '17 at 3:05
• You can still take DE by itself, you just won't know as much context. You will do fine. – kcrisman Jan 30 '17 at 12:14

I'll give you the big picture of the differential equations course I've taught for a few years, I'm not sure what the curriculum is at your university, but, odds are we have much in common.

First, our general goal is to solve differential equations. Depending on the problem, that may mean we seek a function whose derivatives satisfy the given DEqn, but, it may also mean we are happy to find an equation whose differential consequence is the DEqn. For example, $\frac{dy}{dx} = y$ has explicit solutions $y=ce^x$, whereas $(6x^5+2xy^2)dx+(5y^4+2x^2y)dx$ has implicit solutions in the form of the level curves $x^6+x^2y^2+y^5=c$. So, in the implicit case it's not practical to solve for $y$ as a function of $x$.

Writing a differential equation in the form $Mdx+Ndy=0$ is known as Pfaffian Form. Pfaff, a teacher of Gauss, pioneered studying differential equations as equations in differentials. There is a theorem of Pfaff which states that many DEqns of the form $Mdx+Ndy=0$ can be multiplied by $I$ such that $IMdx+INdy = dF$ where $dF = F_xdx+F_ydy$ is the total differential you likely learned in multivariate calculus. Well, this is great, because $IMdx+INdy=0$ then has $F(x,y)=c$ as a solution. Let me give you a specific instance of this, $$\frac{dy}{dx}+py = q$$ is the linear first order differential equation in standard form. We can write that also as: $$dy = (q-py)dx \ \ \Rightarrow \ \ (py-q)dx+dy =0$$ this is not an exact equation because we cannot find $F=F(x,y)$ for which $dF= (py-q)dx+dy$. However, if we multiply by $I = I(x)$ then $$I(py-q)dx+Idy = 0$$ is exact provided $\partial_x I = \partial_y (Ipy-Iq)$ or, $$\frac{dI}{dx} = Ip \ \ \Rightarrow \ \ \frac{dI}{I} = pdx \ \ \Rightarrow \ \ I = \exp \left( \int pdx \right).$$ Perhaps you've seen this as a device to solve $\frac{dy}{dx}+py=q$. Anyway, my point here is that the term integrating factor is used for studying exact equations and also for linear 1st-order DEs and at first glance these techinques do not seem common. Yet, the linear problem's integrating factor is actually a particular instance of the more general idea.

Ok, so, for first order problems, mostly the game is to either use calculus or algebra to wrangle the problem into something we can just integrate to see the solution. This is no longer true for higher order problems. For example,

1. to solve the $n$-th order homogeneous differential equation with constant coefficients we write down its characteristic polynomial, factor it, and read off the solution (this should be derived and explained, but at the end of it we have a simple prescription for solving the DE with not need for integration). For example, to solve $y''+3y'+2y=0$ write $r^2+3r+2 = (r+1)(r+2)=0$ hence $y=c_1e^{-x}+c_2e^{-2x}$ (solved, almost too easy, yes?)

2. to solve the $n$-th order nonhomogeneous differential equation we take the fundamental solution set of the corresponding homogeneous DEqn and run it through the variation of parameters technique.

3. to find that fundamental solution set of the $n$-th order homogeneous problem with possibly nonconstant coefficient functions we use power series techniques (however, in my experience, no one has patience to teach more than second order problems here)

4. To solve systems of differential equations we really need some insights from linear algebra, but, in a nutshell, the matrix exponential is where it's at.

5. Then, there are DEqns with discontinuous forcing functions, for these you want Laplace transforms.

6. Differential equations with singular points (say the regular type), then the Frobenius method is the way to go (this leads to many of the important special functions of mathematical physics)

I have bad news, we're still not to physics. Most of physics is PDEs (partial differential equations, this means more than one independent variable, like space and time in Maxwell's Equations or your namesake's equation). That's a whole other story. But, to understand it, you need a firm grounding in ordinary DEqns. I recommend Habermann's PDE book to get a computational grounding there. For DEqns, it's nice to have a copy of Nagle Saff and Snider's text, get the 3rd or 4th edition, they're inexpensive and have more to say than more recent editions.

In short, the labels in DEqn have to be more sophisticated than in calculus, because, we have so many different kinds of problems to solve. Moreover, as Gerald Edgar indicated, most differential equations we encounter in real world applications cannot be solved via the closed form methods I mention in this answer (or for the 100's of techniques I have not even touched here). Instead, a numerical technique is used to study the DEqns. Simulation plays a big role in what some physicists believe these days. If you can approximately build something on a computer, numerically solving relevant PDEs etc... then many will believe that's a physical possibility.

Definitions "linear", "homogeneous", etc. are things that you need to know to carry out the solutions of these differential equations.

Since most differential equations cannot be solved in closed form, it is useful to know cases that can. (And in the past, a lot of effort has been spent on developing the methods for this.)

You gave one excellent example yourself for knowing correct names for things: Needing to search for the subject later, either online or in an index of some sort.

More generally, be aware that mathematical practice is always a shared endeavor. You need to be able to read, write, and talk to other people about your practice. This may be somewhat masked in the environment up to undergraduate work, where the focus is on being tested in isolation. But along the way you may need to: Read or listen to future presentations, ask questions to professors, understand their answers, work with students in a study group, receive help from a tutor, help another student with a critique or correction, work as a tutor or instructor yourself, read or write a paper discussing new research findings, etc.

You need the shared language to be able to communicate in these contexts. In theory, it might be possible to communicate in writing purely via algebraic expressions; but it would be tedious and fragile and is not commonly done. More pressingly, when speaking verbally it's effectively impossible to precisely express algebraic notation, and you need to know the definitions of the natural-language terminology to make that give-and-take efficient.

One of the important meta-lessons you will learn in differential equations is how to deal with problems you cannot solve completely.

For almost all differential equations, no one knows how to write a closed form formula for the solution. Nevertheless, this doesn't mean you know nothing about the solution. Frequently, one can learn about many properties, either of individual solutions, or about how different solutions are related to each other, without knowing formulas for the solutions. The goal is to extract as much information as possible about the solution even if you cannot solve the equation.

The terms you are learning relate to various properties of the solutions or the relationship between different solutions. Sometimes these properties can be combined with each other to actually solve the equation, but often you won't be able to solve the differential equation, and these properties will still give you useful information.