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Differential Equation Calculus

  1. Jan 8, 2006 #1
    Hello alll. i am enrolled in Differential Equation Calc this semester. i was wondering what kind of stuff will i be doing and what should i expect in terms of difficulty etc. thanks in advance
     
  2. jcsd
  3. Jan 8, 2006 #2
    Do you have a book for the class yet? I reccomend that you take a look at that.

    In general you'll be solving equations of this form:

    [tex]f_n(x)y^{(n)}(x) + f_{n-1}(x)y^{(n-1)}(x) + \cdots + f_0(x) y(x) = g(x)[/tex]

    Where [itex]y^n(x)[/itex] denotes the [itex]n^{th}[/itex] derivative with respect to [itex]x[/itex] of the function [itex]y(x)[/tex]. Don't worry though, you'll start out with simple equations and then work up to the harder ones.
     
  4. Jan 8, 2006 #3

    TD

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    I assume you've had basic algebra so you should be familiar with classical equation in 1 or more variables. I also assume that you've had basic calculus, derivatives and integrals. With differential equations, these two fundamental concepts of calculus meet eachother. In these equations, the unknown is no longer just 'x' (or more variables) but a function y and its derivatives (y', y'', ...).

    You'll be studying (and trying to solve, i.e. integrate) equations which involves derivatives. For example, if y is a function of x, consider one of the most basic DE (differential equations): [itex]y' = y[/itex]. Of course, as you probably know, the exponential function [itex]e^x[/itex] is its own derivative, so a solution of this DE will be [itex]y = e^x[/itex]. Notice that we still have a solution if we multiply this solution with a constant, so the general solution will be [itex]y = ce^x[/itex].

    Now we just 'guessed' the answer, but how could we really find it without guessing? Well you know that y' means the derivative of y with respect to x, y' is nothing more than dy/dx. Solving yields:

    [tex]y' = y \Leftrightarrow \frac{{dy}}
    {{dx}} = y \Leftrightarrow \frac{{dy}}
    {y} = dx \Leftrightarrow \int {\frac{{dy}}
    {y}} = \int {dx} \Leftrightarrow \ln \left| y \right| = x + C \Leftrightarrow e^{\ln \left| y \right|} = e^{x + C} \Leftrightarrow y = ce^x [/tex]

    Now it's also clear why I called 'solving the DE' also 'integrating the DE' and where the connection with integrals lies. It's also easy to see where this constant came from, note that e^C is just another constant which I named c.

    Well this was very basic but I hope now you get the idea about differential equations, we're trying to find a function which satisfies the differential equation for all x :smile:

    Good luck!
     
    Last edited: Jan 8, 2006
  5. Jan 8, 2006 #4
    ah, thanks. The reason i asked is becuase the professor was late in specifying what book we needed so the campus bookstore has yet to order it. Also, The class has an emphasis on engineering and using matlab to solve DE. has anyone done this before? i'm not sure how this can be done because i know matlab is not a symbolic based program (like maple is) and is a decimal based, so does anyone have any insight to how this can be accomplished?
     
  6. Jan 9, 2006 #5

    HallsofIvy

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    I honestly don't know what "Differential Equation Calc" means. Normally you take several courses in calculus before taking "Differential Equations" because you have to really know calculus well before you can understand differential equations. I would have thought an engineering course would do more than just teach you how to use a program to solve differential equations!
     
  7. Jan 9, 2006 #6

    BobG

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    dicerandom's comment reminds me of the nickname our instructor had for DiffEQ. He called the course "Difficult Equations".

    Actually, it's not that bad, however it requires most people's least favorite algebraic technique. Eventually, you use LaPlace Transforms to transform a 'difficult' problem into one that can 'easily' be solved using partial fractions.

    If the course is like most, you could get by without any computer assisted program. They push using a CAS because it's a good idea to start learning how to use Matlab or some other program while still solving problems easy enough to check your results by hand. (Similar to first learning how to use a calculator while the problems are simple enough to solve in your head).
     
  8. Jan 9, 2006 #7
    no, they aren't just teaching us to use a program to solve Differential equations. they still expect us to know how to do it all by hand as well. I think the idea of using Matlab is to enhance our knowlege of DE.

    Also, i have taken two years of calculus prior to this class, so i think im ok in the calc department.

    BobG, thanks for the relpy. i'm pretty good with Matlab, as I had an engineering class that was soley devoted to teaching programming in MAtlab, C++ and html, so im quite confortable with the program
     
  9. Jan 9, 2006 #8

    saltydog

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    I use Mathematica. Suppose you were solving:

    [tex]y^{''}-\frac{1}{Tan[x]}y^{'}=e^x Sin[x][/tex]

    and you obtained (by hand) the solution:

    [tex]y(x)=c_1-c_2Cos[x]+1/2e^x(Sin[x]-Cos[x])[/tex]

    How do you know it's right?

    Well, you can back-substitute it and that's what you need to do once in a while for practice but sometimes your focus is on other matters (like solving a problem) and don't wish to spend time back-substituting but you still want confidence your work is correct so just enter into Mathematica:

    Code (Text):

    [tex]y[x]=c_1-c_2Cos[x]+1/2e^x\left(Sin[x]-Cos[x]\right)[/tex]

    [tex]y^{''}[x]-\frac{1}{Tan[x]}y^{'}[x]==e^x Sin[x]\text{//Simplify}[/tex]
     

    If correct Mathematica returns TRUE. Now I don't know about anyone else but to me that's amazing how that one command (the second one) did all the differentiation without me telling it explicitly to do so, made the substitutions, did all the arithmetic, and then checked the results.:smile:

    There are all kinds of other similar scenarios which allow Mathematica, used in this way, to be a great teaching tool as long as one remembers not to do anything with it that you can't already do (usually) by hand.:smile:
     
    Last edited: Jan 9, 2006
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