Re-studying Series for ODE's due to weakness in material

[PhotonSSBM]

My series foundation is really weak, and this spring I'll be taking Differential Equations. I know series plays a big role in solving some ODEs, so I'll be re-learning the material from Calc 2 to make sure I'm up to par for the class. I'll be working through Stewart's chapter on series but was wondering what I should prioritize and if maybe I could leave some stuff out. Any supplemental material suggestions would be appreciated.

 

[MidgetDwarf]

Hmm, my understanding of ODE intro courses, is that they are recipe courses in general.

You will have to know how to add Summations, whether or not the series converges, radius and interval of converges ,and taylor series expansion about x=0 (McLaurian Series) of the basic functions.

They where all covered in my ODE textbook. However, depending on the rigor of the problems, you may need to review a Calculus 2 book (writing the terms of a series in summation notation).

What i would focus on more is on integration techniques. I would recommend and older edition of Zill. It is a very cook-book, however you can supplement it with say, Coddington. I purchased an older edition for 5 bucks. I swear, that my 8th ed with modeling functions is the same as the newer 8th ed with boundary problems. The difference in price was 120 dollars.

 

[micromass]

My series foundation is really weak, and this spring I'll be taking Differential Equations. I know series plays a big role in solving some ODEs, so I'll be re-learning the material from Calc 2 to make sure I'm up to par for the class. I'll be working through Stewart's chapter on series but was wondering what I should prioritize and if maybe I could leave some stuff out. Any supplemental material suggestions would be appreciated.

Depending on the class, you might want to review some linear algebra too. Maybe the class won't require it though.

 

[Remixex]

Read the program of the course if you can, ODEs that solve with power series (which are non-linear, i believe) are not something of interest in many Differential Equations courses.
If you posted a list of the general subjects the course will touch upon the advice might be better, something important though, integration has to be a natural for you, if you still are weak in that area i suggest studying that over series, because that is used in almost EVERY god damn differential equation, but power or infinite series are only used in specific ones.

 

[S.G. Janssens]

ODEs that solve with power series (which are non-linear, i believe) are not something of interest in many Differential Equations courses.

What do you mean to be non-linear? The ODE or the series? And in the latter case, what do you mean by that?

Power series solutions for linear ODE with non-constant coefficients are quite important in mathematical physics. They lead to linear recursions. Trying to solve a nonlinear ODE by substituting a power series would probably lead to a nonlinear recursion. I have never seen this being used effectively.

god damn differential equation

"God made the integers, all else is the work of man." (L. Kronecker)

 

[Remixex]

I'm an undergrad with only 1 course of DE on my back so what i said is most likely wrong,i get confused between linear and non-constant coefficients, i meant the latter.
I've never had to solve an ODE with power series on my own because the approach of the course i took was to see only basic oscillations of 1 variable and then it deviated into PDEs with separation of variables and Sturm-Liouville problems.

Power series solutions for linear ODE with non-constant coefficients

Exactly, we only saw constant coefficients, so i wouldn't know about that (the only non-constant coefficient I've seen is an Euler equation)
After a while i saw the Bessel equations, i believe those need series to solve, but it might be too far fetched for the first course he takes.
That's why I'm asking him to tell me the program of the course, if it's basic and oscillations-driven like mine, he might never see variable coefficients until he goes deeper into the subject.
Granted, we did solve some equations through Fourier, but they had constant coefficients.

 

[PhotonSSBM]

Hmm, my understanding of ODE intro courses, is that they are recipe courses in general.

You will have to know how to add Summations, whether or not the series converges, radius and interval of converges ,and taylor series expansion about x=0 (McLaurian Series) of the basic functions.

They where all covered in my ODE textbook. However, depending on the rigor of the problems, you may need to review a Calculus 2 book (writing the terms of a series in summation notation).

What i would focus on more is on integration techniques. I would recommend and older edition of Zill. It is a very cook-book, however you can supplement it with say, Coddington. I purchased an older edition for 5 bucks. I swear, that my 8th ed with modeling functions is the same as the newer 8th ed with boundary problems. The difference in price was 120 dollars.

Thanks for the useful post. My integration skills are sharp from doing complicated double and triple integrals all semester. So I should be fine there.

Depending on the class, you might want to review some linear algebra too. Maybe the class won't require it though.

The class doesn't require it as a prerequisite. But on the off chance it comes up what areas of LA will be important?

 

[micromass]

The class doesn't require it as a prerequisite. But on the off chance it comes up what areas of LA will be important?

It really depends on the course. But LA will be important if you study linear systems of linear equations, and linear equations of higher order. What is important there are determinant, linear dependence, vector spaces, subspaces. Those will get you pretty far. But if LA is not a requirement, then this will probably not be covered.

 

[PhotonSSBM]

It really depends on the course. But LA will be important if you study linear systems of linear equations, and linear equations of higher order. What is important there are determinant, linear dependence, vector spaces, subspaces. Those will get you pretty far. But if LA is not a requirement, then this will probably not be covered.

Cool. I'm good on all those fronts so even if it comes up I'll be ok. Thanks :D