Date: Mon, 15 Oct 2001 20:57:17 -0500 (CDT)
To: organism
Subject: Plea for help!

We're a group in 427K. After attempting to solve the equations for the group
assignment using Maple, Mathematica, and Matlab, we've ran into problems. All of
the programs are spitting out huge, page-length, equations, and foreign matter.
(What's with the funky ae in Mathematica?!?)  We've spent a lot of time trying
to figure these problems out. We were wondering if we were doing something
wrong. Here's what we have entered, any help would be appreciated:

Mathematica:
p. 178
15) DSolve [y''[x] - 2y'[x] + y[x] == x*e^x + 4, y[x], x]
DSolve [{y''[x] - 2y'[x] + y[x] == x*e^x + 4, y[0] == 1, y'[0] == 1}, y[x], x]
18) DSolve [y''[x] + 2y'[x] + 5y[x] == 4e^(-x) * Cos(2x), y[x], x]
DSolve [{y''[x] + 2y'[x] + 5y[x] == 4e^(-x) * Cos(2x), y[0] == 1, 
    y'[0] == 0}, y[x], x]

Maple:
p. 178 15)
diff_eq1 := D(D(y))(x) - 2*D(y)(x) + y(x) = x * e^x + 4;
init_con := y(0)=1, D(y)(0)=1;
dsolve ({diff_eq1, init_con} , {y(x)});
simplify (dsolve ({diff_eq1, init_con} , {y(x)}));
dsolve ({diff_eq1} , {y(x)});


ALSO, for p. 184, #5, 6, & 7,  how do we enter the inequalities as initial
conditions? Thanks again!


From organism Tue Oct 16 12:23:35 2001
Date: Tue, 16 Oct 2001 12:22:53 -0500 (CDT)
From: jr <organism>
Subject: Re: Plea for help!

> All of
> the programs are spitting out huge, page-length, equations, and foreign matter.

Ah, the joys of using computer programs to solve your problems! ;)
Remember that the problem set asks you to enumerate some of the
disadvantages of using these programs.

> (What's with the funky ae in Mathematica?!?)

What's "ae"?

> We've spent a lot of time trying
> to figure these problems out. We were wondering if we were doing something
> wrong. Here's what we have entered, any help would be appreciated:
>
> Mathematica:
> p. 178
> 15) DSolve [y''[x] - 2y'[x] + y[x] == x*e^x + 4, y[x], x]
> DSolve [{y''[x] - 2y'[x] + y[x] == x*e^x + 4, y[0] == 1, y'[0] == 1}, y[x], x]
> 18) DSolve [y''[x] + 2y'[x] + 5y[x] == 4e^(-x) * Cos(2x), y[x], x]
> DSolve [{y''[x] + 2y'[x] + 5y[x] == 4e^(-x) * Cos(2x), y[0] == 1,
>     y'[0] == 0}, y[x], x]

In Mathematica, the base of the natural logarithm is denoted by "E" and
not "e".  Likewise, the imaginary unit is "I" and not "i" (not that you'd
use it anyway).  You can also substitute "Exp[x]" for "E^x".  You're
correct in capitalizing "Cos" but remember that all function calls in
Mathematica use brackets - "[" and "]" - and not parentheses to set off
arguments.  Thus you want "Cos[2x]" and not "Cos(2x)".

Also, remember that you can use Mathematica (and Maple) to just help you
in a particular step of a problem solution, and not just in solving the
whole problem for you.  For instance, it could help you solve the linear
system of equations that result from the method of undetermined
coefficients.  It could solve for the homogeneous solution for you.  It
could solve the system of equations that result from incorporating the
initial conditions.

> Maple:
> p. 178 15)
> diff_eq1 := D(D(y))(x) - 2*D(y)(x) + y(x) = x * e^x + 4;
> init_con := y(0)=1, D(y)(0)=1;
> dsolve ({diff_eq1, init_con} , {y(x)});
> simplify (dsolve ({diff_eq1, init_con} , {y(x)}));
> dsolve ({diff_eq1} , {y(x)});

I'm not as familiar with Maple, but I'd guess that there are syntax
problems here.  For instance, try "exp(x)" instead of "e^x".  I'd guess
that "D(y)(0)=1" isn't legal syntax either (as well as "D(y)(x)" - but
then again, maybe you're right).  Check the documentation that comes
built-in to Maple; it should have examples.  Failing that, ask the lab
proctor for help.

> ALSO, for p. 184, #5, 6, & 7,  how do we enter the inequalities as initial
> conditions? Thanks again!

These are things which can be difficult for Mathematica and Maple to
represent.  One way around this is to consider just why those intervals
are specified: the right-hand sides of those ODEs have discontinuities at
the endpoints of the given intervals.  We know from the existence and
uniqueness theorem that we can't expect a solution to exist (or to be
unique if it does) past the times where the RHS (or the coefficients of
y, y', etc.) become discontinuous.  Stated another way, if we're given
initial conditions at a particular time, we can only integrate the DE to
get a solution up to where the RHS (or the coefficients) are
discontinuous.  Thus, tell Mathematica (or Maple) that your initial
condition is somewhere in the interval the book specifies.  For instance,
in problem #5 you could try:

DSolve[ {y''[t]+y[t]==Tan[t], y[Pi/4]==a, y'[Pi/4]==b}, y[t], t ]

in Mathematica.  Note that I haven't specified "a" or "b".  Mathematica
and Maple can give symbolic solutions in this instance.  (You'll need to
play with this, though, to get it to work correctly.)

Again, another thing to consider is that you don't have to try to get
these programs to solve the whole problem for you.  Use the programs to
solve the homogeneous equation, and then calculate the integrals which
result in the variation of parameters method.

Hope this helps!

j