Why am I struggling with Differential Equations?

[andryd9]

Why am I struggling with Differential Equations??

Please help: I did well in Calc I-III, and now am struggling in Diff.Eq. Anyone else find themselves in the same situation, and how did you save yourself? TIA:)

 

[mathwonk]

is it bad to struggle? apparently it is new.

 

[andryd9]

Yes, it is bad. And yes, it is new. If I am hindered in my progress and can't get a fix on how to adapt, I've wasted my time in this course. I was hoping to hear constructive advice from others who experienced the same. No guarantee, I know, that it will apply to my particular case, but one can always try.

 

[hadsed]

How can you expect constructive advice when the explanation for your situation is literally one sentence? Maybe you should expound a bit on what you're facing, why you feel like you're struggling, what exactly you're struggling with, etc.

 

[andryd9]

Ah, but there is a commonality of struggle:) Most students find the same things challenging in the same courses. I was hoping to hear the perspective of others who mastered diff.eq with effort. How about you- how did you find it comparable to your calc series? Were you well-prepared? Looking back, would you have preferred a different text? No intent to be cryptic, but others' hindsight about their own experiences can inspire a wealth of advice- perhaps better than if I solicited info specific to myself.

 

[Jorriss]

Ah, but there is a commonality of struggle:) Most students find the same things challenging in the same courses. I was hoping to hear the perspective of others who mastered diff.eq with effort. How about you- how did you find it comparable to your calc series? Were you well-prepared? Looking back, would you have preferred a different text? No intent to be cryptic, but others' hindsight about their own experiences can inspire a wealth of advice- perhaps better than if I solicited info specific to myself.

I think the best differential equations book is Ross, Differential Equations.

There are common themes of what give people trouble but it's still helpful to know what is hindering you.

 

[Poopsilon]

The only struggle for me in diffyQ was staying awake. The course is purely algorithmic, decide on the correct recipe to apply and watch your sign errors.

 

[ZenOne]

^^What Poopsilon said...^^

 

[Meadman23]

What helped me a lot was Schaum's Outline of Diff Eq (I used the 3rd Edition). With the book we were using, it would often just present the theory, but not an example of the theory in action. What also helped me was to understand how to identify basic forms of differential equations such seperable, homogenous, etc.

I'll also add that cramster helped loads to. Many times you might get the right answer, but be unsure of it was coincidence or the correct method. Using cramster, I could follow the steps and see if they matched up with mine.

 

[Mmm_Pasta]

I'll also add that cramster helped loads to. Many times you might get the right answer, but be unsure of it was coincidence or the correct method. Using cramster, I could follow the steps and see if they matched up with mine.

There isn't always one way to get the correct answer. Also, I have seen quite a bit of mistakes on that site.

 

[bpatrick]

I've never known anybody having troubles in an ODE course if they were adequately prepared for it, it's pretty cut and dry algorithmic solutions and whatnot ... and if you did fine in calc 1-3 not sure what the problem would be ... linear algebra maybe? If that's the problem, then just start reviewing the most important computational sections of LA for ODEs (determinants, wronskian, eigenvalues/vectors).

I guess some profs may put some 1st order problems on their exams that involve solving tricky integrals ... if that has been a problem, spend time reviewing integration by parts, trig substitutions, and partial fractions decomposition.

Other than that, my best advice would be to make an outline for yourself ... maybe a page long on the categories of DEs (with examples) you're dealing with and then their solution methods. Examples:

x^2y'=Ay >>>>>>>> separate to get x's on one side and y's on the other then integrate and solve for y.

Ay'' + By' + Cy = 0 >>>>>>>>>> solve Ad^2 + Bd + C = 0 for d by factoring or quadratic formula, then putting the solutions into y=(c1)e^(d1)x + (c2)e^(d2)x ... or making whatever adjustments you need (like repeated roots, converting e^(a+bi) stuff into sin/cos, etc...)

Ax^2y'' + Bxy' + Cy = 0 (cauchy-euler) >>>>>>> solved by y=x^p ... substituting, and solving for your p values.

so yeah just do that for everything you've learned so you have a cheat sheet to use until it's all second nature.

to make it second nature, just practice loads of problems, and try to do it randomly (possibly by having a friend pick problems for you from various sections of your book and writing them down for you to solve in a jumbled order), so you won't already know the solution method ahead of time due to the section of the book you're in (since all the problems in section 3.2 or whatever are the same solution method = just repeating the same thing over and over rather than having to analyze which solution method to use like you'll have to do on exams).

 

[Meadman23]

There isn't always one way to get the correct answer. Also, I have seen quite a bit of mistakes on that site.

It's true there is more than one way and there may be errors there, but I still maintain the sites helpfulness.

 

[Jorriss]

I thought ODEs wasn't too bad but there were some tricky problems in the class. One of the problems was taking this higher order differential equation and turning into a 4 x 4 system of differential equations. That was tricky.

 

[psparky]

Diff EQ is one of your tougher courses in engineering.

You are suppose to be struggling, therefore you are right on track. Congratulations.

The only thing that is going to help you...is hours and hours and hours of studying. There is no shortcut. Eventually, the bell will ring and you will pass a test. Keep in mind too...your fellow students are also struggling...you are not alone!

 

[ZenOne]

I disagree--ODE is NOT one of the harder courses in engineering--did you have to do dynamics? Mass and heat transfer? Mechanical vibrations? ODE is a joke compared to those--IMO, of course.

Hell--calculus 3 (for engineers, with all the application problems) was harder than ODE.

 

[psparky]

I disagree--ODE is NOT one of the harder courses in engineering--did you have to do dynamics? Mass and heat transfer? Mechanical vibrations? ODE is a joke compared to those--IMO, of course.

Hell--calculus 3 (for engineers, with all the application problems) was harder than ODE.

Difference of opinion. I thought dynamics was simple...and I thought calc 3 was a cake walk because it was identical to calc II with just the extra z dimension that followed the same rules. Didn't take the other two because I'm an EE.

Diff EQ is just tough because it's a different way of thinking.

Just my opinion though.

 

[ZenOne]

I agree Dynamics is not the hardest but it is harder than ODE simply because it involves thinking--ODE does not IMO. As someone pointed out--it's a cookbook type class. Oh--Bernoulli--apply method. Oh, Cauchy-Euler--apply method.

Like you said--difference of opinion--I respect that.

 

[dijkarte]

I blame the lecturer :D

Well if you are taking a course and you usually do or did well in other courses even less advanced, then I definitely blame the instructor. I had a bad experience with some of the most beautiful topics in math to find out later it was not my fault or any student at the same class. It was the teacher who deliberately succeeded in making us hate the subject at that time. And he was enjoying it. It's not about bad teaching skills, it comes to the fact that some uni's try to make many student re-take the course so they can upgrade their marks., so simply it's $$$ matters. However I've encountered teachers with very bad teaching skills so we had to do much work on our end, but we did well because the teacher's intention was not to screw up students for whatever psycho-economic reasons. When the instructor lacks the teaching skills, then I'm with the argument that get a good mark and do it later yourself.

 

[SophusLies]

is it bad to struggle? apparently it is new.

lol, yeah. I struggled in almost every class I ever took in college because I specifically took the hardest courses my adviser would let me. If you're not struggling then you're not learning anything.

Were you well-prepared?

Not really, I self studied some things the summer before I took it.

Looking back, would you have preferred a different text?

I didn't mind the text although I would have preferred more insights into physics because DE's and physics go together so well but it is a math class. If I ever had a problem with the textbook I would be in the library searching for another book that would better explain whatever I was struggling with.

 

[mathwonk]

I also struggled in an algorithmic ode class as my first math course after being out of school a year. I began supplementing the course with a schaum's outline series, and also going to the library for an hour after every class to review the notes. Finally at the end the class got less routine when the prof gave a proof of existence of solutions by the beautiful contraction mapping method. That even made it interesting. I went from a D to an A.

Then when I taught it I tried to use more interesting books, like Martin Braun's well written book, supplemented by V. Arnol'd's book. The standard books like Boyce and diPrima really left me cold. Almost everyone agrees that he best, clearest book is probably the following one by Tenenbaum and Pollard. Try that one.

https://www.amazon.com/dp/0486649407/?tag=pfamazon01-20Here is my review:

10 of 10 people found the following review helpful
5.0 out of 5 stars unique, March 28, 2006
By mathwonk - See all my reviews
This review is from: Ordinary Differential Equations (Dover Books on Mathematics) (Paperback)
i discovered a "new" method of solving constant coeff linear ode's this semester while teaching the course, No one I asked knew it and no books had it, but it was so natural as to have no chance of being really new. Still I searched and searched, Courant, Loomis and Sternberg, Edwards Penney, Coddington, Braun, Dieudonne, without success. then I found it here on pages 268-292.

i was impressed. this book was written back when clarity and completeness were the goal. then i began looking at the problems. it is very hard to give reasonable example problems using variation of parameters that cannot be solved better by guessing, but tenenbauim and pollard do it.

this is a classic introductory text. they even define differentials correctly, almost unheard of in an elementary book. all this for only 16 bucks!

 

[Angry Citizen]

I agree Dynamics is not the hardest but it is harder than ODE simply because it involves thinking--ODE does not IMO. As someone pointed out--it's a cookbook type class. Oh--Bernoulli--apply method. Oh, Cauchy-Euler--apply method.

Like you said--difference of opinion--I respect that.

That's the thing, I think. I (and many other engineering students that I know) prefer a dynamic (read: malleable) problem-solving process, not an algorithmic one. I'm at my best when all I need to know is where to start on deriving a particular formula (for instance, conservation of linear momentum starting from F=ma). Dynamics (read: the course) is quite an easy class compared to something like ODE's, which requires memorization of the algorithm and a perfect knowledge on how to solve complex integrals by hand - a relatively useless talent, in my opinion.

 

[psparky]

Like others said...I think Difff EQ is tough just because it deals with things in a way you've never seen before...a different way of thinking.

Dynamics is simply the same stuff you saw from physics and statics...the material may be new...but the way of thinking is quite similiar.

 

[kjohnson]

Differential equations isn't necessarily a "cookbook" class, it just depends on how its studied/who is teaching it. Yes, in many undergrad engineering classes it is taught in a cookbook style, but just because you know a few tools to solve a limited set of problems doesn't mean you understand what's really going on. Considering things like uniqueness,stability,etc...you can get pretty involved if you really want.

Likewise different teachers may take a more rigorous approach and thus it may be initially harder to grasp. So saying that differential equations was a cakewalk class is not really fair to people studying the subject currently.

 

[mathwonk]

imagine a plane with arrows drawn beginning at every point. this is a differential equation. There are two kinds of solutions.
1) velocity vectors. this approach looks at the arrows as velocity vectors for a moving particle (to be found). thus a solution would be a parametrized curve in the plane such that at every point its velocity vector matches the arrow drawn at the at point.

this type of solution only "fills up" a curve of arrows, so a full solution would be a family of such curves, one through every point. These do tend to exist.

2) gradient vectors. In this approach we seek a "potential" function, i.e. scalar function in the plane whose gradient equals the given arrow at every point. This sort of solution would solve the equation at every point, but these solutions tend not to exist. they exist only for "exact" vector fields, i.e. ones with no "curl".

For some reason standard books i learned from never explained these simple descriptions of what the geometry of an ode is about.

 

[Angry Citizen]

imagine a plane with arrows drawn beginning at every point. this is a differential equation. There are two kinds of solutions.
1) velocity vectors. this approach looks at the arrows as velocity vectors for a moving particle (to be found). thus a solution would be a parametrized curve in the plane such that at every point its velocity vector matches the arrow drawn at the at point.

this type of solution only "fills up" a curve of arrows, so a full solution would be a family of such curves, one through every point. These do tend to exist.

2) gradient vectors. In this approach we seek a "potential" function, i.e. scalar function in the plane whose gradient equals the given arrow at every point. This sort of solution would solve the equation at every point, but these solutions tend not to exist. they exist only for "exact" vector fields, i.e. ones with no "curl".

For some reason standard books i learned from never explained these simple descriptions of what the geometry of an ode is about.

Erm.. no. Can't say I ever got that version either. Thanks for the informative post.

 

[psparky]

In electrical (and other disciplines as well)...we get "s domain" transfer functions from certain circuits...usually controls...such as 1/((s+10)*(s+100))...

Expand it out and you get s^2 + 110s + 1000...also known as s" + 110s' +1000...in other words it's a differential equation. A 2 pole situation that is non linear. We can also get a "bode plot" from this (graph that shows gain at every frequency...not to mention phase shift). "Jw" can also be substituted for "s" when looking at bode plots. Plug in your "w" (omega) and you will get the exact mathmetical gain and frequency shift...at that specific frequency...(or radians)

I don't know the entire big picture...but it is kinda neat how it all ties together.
Hopefully, someone can expand upon what I just said...or correct me...or whatever.

 

[Alesak]

There is this book that explores connection between Lie groups and PDEs. Perhaps it is more systematic and "why" approach to DEs that the usual cookbook method? I have no idea what is this book about, would be grateful if anyone could comment.

 

[homeomorphic]

I got an A in calc 3, but a C in diff eq. It was before I was very good at math, but still, a lot of the problem was that we used Boyce and DiPrima, which doesn't convey much of an understanding of the subject.

 

[zif.]

I struggled with ODEs just because I didn't spend enough time with it.

I managed to make everything click in the end (about a day before the final and therefore salvage an ok grade) and it was one thing that helped:

Make a flowchart for the different types of problems and their different subsets and then memorize it.

In our second year ODE class there really was only like 9 different types of problems.

Then just pick different problems from the relevant sections of any textbooks, and do them until you know all the steps.

As it was said before, there's very little understanding actually required for this course, you just have to memorize the steps.

 

[andryd9]

Thanks everyone who replied. I considered all practical advice, (esp. about reviewing and textbooks) and ended up doing fine. Thanks again:)

 

[Skrew]

My problem with my first ODE class was that theory was ignored for a lot of it.

Maybe that's the case with your class.

 

[RoshanBBQ]

Here's my advice:

1.) You can use MIT lectures. There is one for differential equations. http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/ Heck, there are also some assignments and solutions. See if you are on par with an MIT student!

2.) I found that signals and systems helped piece differential equations together for me a bit. You might want to consider looking into buying a good signals and systems /controls book as a supplement. Its entire mission is to approach differential equations in the most manageable style possible.

 

[20Tauri]

I did fine in calculus, and I'm taking Differential Equations now. Our book is pretty terrible--it's Borelli and Coleman, I think? I do okay solving the equations, but I am confused a lot of the time. I think about half my problems are due to not really understanding the theory behind what we're doing, and half due to forgetting things from linear algebra and calculus, because I took those a while ago. Going over the book and the notes helps. If there's someone in your class who understands the material better than you do, it might be helpful to do homework together and talk through it.