Differential equationshow difficult?

[Coolphreak]

How difficult did most of you find the first course in differential equations? What math background did you have when you first took it? I have taken Math until calc 2 and linear algebra. Do a lot of solution methods involve series?

 

[ranger]

I found ODE to be quite straight forward. You will need your integration skills a whole lot. So I would suggest you revise your calc II material. When I took DE, I had only taken up to calc II and linear algebra. You should be absolutely fine, providing that you're integration skills are good.

 

[symbolipoint]

The study of differential equations, if it is packaged with the combination "Introductory Differential Equations and Linear Algebra" course, is very easy. You will only study a few simple kinds of differential equations and applications. This may vary depending on the school and who teaches the course. That combination course is one typical of most undergraduate physical science students who only need 3 or 4 semesters of Calculus. For other majors and intents, what I have just said might not apply. The dedicated Differential Equations courses probably are much more thorough and tougher (just a guess).

 

[mathwonk]

differential equations in general are extremely difficult to solve. that's why first courses focus on the only easy cases, exact equations, especially first order, and linear constant coefficient case.

the constant coefficient case is the easiest becaUSE THERE THEY BEhave almost exactly like algebraic equations.

i.e. just aS YOU solve an algebraic equation like X^2 -3X+2 by factoring into (X-2)(X-1), so also you solove the de y'' - 3y' +2y by factoring the operator D^2 -3D+2 as (D-2)(D-1).

i.e. if you can solve y'-2y = f, and y'-y = f (which reduces to an exact equation by the first order theory) yhen you can also solve their "product" y''-3y'+2y=f.

have fun.

remember: learn to solve first order equations by making them exact via an integrating factor. then learn to solve higher degree constant coeff cases by factoring them and applying the first r9der acse inductively.

thats all there is a first course.

(the reason nonm constant coeff cases re harder is the operators do not commute.)

oh yes, there are soime other methods involving expanding the given functions as sin and cosine series or powers eries, and then giving infinite expresions for the solkutions in terms of the known integrals of those simple functions. these are called powers eroes methods and Fourier series methods, also some people like laplace transforms in the same way some people used to emply log tables to do multiplication problems.

non explicit geometric methods are most powerful, such as in blanchard and devaney, or as much better treated in arnol'd.

 

[ekrim]

The first course in differential equations I took was pretty easy. There were less topics to remember than in the calculi, and the subject matter (at least at my school) was more computational, rather than theoretical. I pretty much put in as little effort as possible to get an A, which is a big regret of mine as i have forgotten everything

 

[^_^physicist]

Differential Equations was the first math class I took at university (with no college calculus), and I got an A. It isn't tremendously difficult if you take the introduction courses; however, other courses that have had differential equations in them have been difficult; however, they weren't impossible with a bit of practice.

Mathwonk is entirely correct, Differential equations on a whole are very difficult to solve; yet, introduction courses give you a small varitiy for methods which are realitivly easy to use.

Series can be useful for some courses, but it largely depends on the school.

Best of luck.

 

[arunma]

Slightly off topic, but I would recommend that all the physics undergrads pay careful attention to their differential equations course. Just today I took my classical physics PhD qualifier, and there were two problems which required heavy use of differential equations. Fortunately I was a math major in college, and so I was able to solve them pretty easily (they were not unlike what you might find in a sophomore-level course). In fact, some of the tricks from my second-year undergrad course came in handy. However, some of my fellow students said that they were unable to solve these problems. So it turns out that this stuff is really important. You never know when you might be taking a physics test which will determine the future of your academic career, and suddenly a non-linear but separable differential equation will show up!

 

[mathwonk]

which tricks did you use?

 

[matt grime]

Differential equations as a theory is *hard*. Differential equations as a mathematics class is *easy*, since you will only be given questions for which you have the tools required. So, learn the tools and do it.

 

[arunma]

which tricks did you use?

The tricks I used were pretty simple, but they were the kind of thing that a physics person without a math background might not get. For example, in one equation the square of the function appeared, but with a bit of algebra the equation became variables separable, and could be solved by integration. On another one it was obvious that there were exponential, sine, and cosine terms, but I needed to guess the coefficients on the trigonometric functions. It all seemed easy to me, but at least two people I talked to said they weren't able to do it. I guess the problem with physics is that the only differential equation they make us know the solution to is u'' + k²u = 0, and even there we tend to just memorize it.

 

[sam1]

Your first course in ODE should be pretty straightforward. I'm in engineering, so the course was basically all problem solving/applications, with little to no existence/uniqueness theorems/proofs required. The entire class was basically "here's an equation, solve it." My intro ODE course covered:

- first order differential equations (separable, linear, exact, substitutions)
- higher order DEs (reduction of order, linear eqns with constant/variable coeffecients, superposition, annihilators, variation of parameters, cauchy-euler eqns)
- intro to nonlinear eqns
- solutions by power series (around ordinary/singular points, touched on bessel functions, legendre polynomials)
- laplace transforms
- systems of ODEs (linear first order only)