Maple is a well-known computer algebra system and it has something called dsolve which produces
solutions to initial value problems. The following illustrates what you type in to have it do this for you.
Before typing in the syntax below, you need to go to file and new and then workshop mode. If you click on
“text” in the upper left corner, what you type appears in red. If you click on math, it will be in italics and
will appear in standard math notation. If you want the commands to appear on new lines, youdo shift enter to get this to happen. However, there is no harm in using a single line.
Since you have selected workshop mode, you will see a red > on the left. After this, type the
following:

Now you press “enter”. The letters “rhs” are used to assign the value of the right side to the left. Then it
will solve the equation and also give a graph of the solution. If you look at what you typed, you will see
how to adjust the length of the interval on which the graph will be drawn. Here is what results from the
above.

PICT

It also gives the solution is y =

1+34e−4x

and the graph of the solution is shown for x ∈ [0,2]. In what
you typed above, diff(y(x),x) signifies y^{′}

(x)

. The rest should be pretty self explanatory. You can change
the differential equation if you want. If you want to write in y^{′′}, this is signified by diff(y(x),x$2), and
I think you can see what it would be for y^{′′′},y^{(4)
} and so forth. In principle, you can do various kinds of
differential equations this way. Of course the problem is that neither you nor the computer algebra system
may be able to find a simple analytic solution. In this case, there may still be a solution, but you
won’t have an answer for it in terms of known elementary functions. Maple can do this case
also.

Before showing the syntax for obtaining a numerical solution with Maple, consider the above claim that
even though you might not have a solution in terms of a simple formula, you might still have a solution.
Say you have y^{′} = f

(t,y)

,y

(0)

= y_{0} and you want the solution on some interval

[0,a]

. One way to
obtain an approximate solution would be to replace the derivative with a difference quotient as
follows:

yi −-yi−-1= f (ti,yi),i ≥ 1
h

where here you have a uniform partition of

[0,a]

,t_{0}< t_{1}<

⋅⋅⋅

< t_{n} = a where h = t_{i}−t_{i−1}. I will leave it
to you to see that there is a solution to the above discrete problem. Then if you want, you could define a
function y_{n}

(t)

to be a piecewise linear function which equals y_{i} at t_{i}. This is an example of a numerical
solution. The above method is called the Euler method. It isn’t as good as what Maple will
use.

Suppose you wanted to use Maple to solve the initial value problem.

′ ( 5)
y = − y+ .01y , y(0) = 1

Then you would type in the following after the red >. Remember, if you want the commands on new lines,
you press shift enter.

with (plots):
de:=diff(y(x),x)=-(y(x)+.01*y(x)ˆ5);
Y:=dsolve({de,y(0)=1},y(x), numeric);
odeplot(Y,[x,y(x )],0..5);

Note the top line which says with(plots): You have to include this so it knows how to plot the solution
you are going to get. Then after typing in the above stuff, you press return. What you get is a graph of this
solution.