> 
# A brief introduction to Differential Equations in Maple.
# Examples include: dsolve, odeplot, and DEplot1.
# 
# Written by Doug Hensley
# 
# DIFFERENTIAL EQUATIONS
# 
# The boldface regions are executable code. Position the cursor inside such a region, 
# click, and hit the button marked Return. Output should appear.
> 7*8;

                                       56
> 
# You can try out your own variants on what you've just seen. Just postion the cursor 
# to the right of a lone boldface prompt character (>),  click, and type in your variant. 
# Remember you need a semicolon at the end. Then hit Return as before. 
> 
> 
# There are large classes of differential equations which have solutions that can be 
# expressed in terms of elementary functions. Many more differential equations have 
# solutions which cannot be expressed as a finite combination of the so-called 
# elementary functions. 
# 
# When such a solution does exist, Maple can often find it. The command which tells 
# Maple to attempt such a solution is "dsolve". For instance, suppose we're really lazy 
# and we want Maple to solve the differential equation y'(x)=x. The code to key in 
# would be:
> dsolve(diff(y(x),x)=x,y(x));

                                           2
                               y(x) = 1/2 x  + _C1
--------------------------------------------------------------------------------
# There is an arbitrary constant, with the peculiar name "_C1", in the answer.  Except 
# perhaps for the terminology, you probably already knew all this. It is just another way 
# of saying that the indefinite integral of x is x^2/2+C. Maple uses the odd name to 
# avoid collisions with variables you may introduce.
# 
# A differential equation becomes more than just alternative terminology as soon as 
# the unknown function itself enters the story. For instance, consider the equation 
# y'(x)=y(x). Again, you probably can think of a function y(x) which has this property. 
# But let's see what Maple does with the task:
# 
> dsolve(diff(y(x),x)=y(x),y(x));

                                y(x) = exp(x) _C1
> 
--------------------------------------------------------------------------------
# 
# Of course this is a solution. But why is the constant now a multiplier, rather than 
# added? How can we be sure there are not others? The answers to these questions are 
# to be found in the course content---one cannot expect the computer to do or know 
# everything!
# 
# As mentioned, many differential equations defy solution in closed form. For 
# instance,  the differential equation y'(x)=x^2+y^2 has no such formula for its 
# solutions. There IS a solution, one for each possible choice of y(0). 
# If you decide you want y(0) to be 1, then you can get Maple to carry out a rather large 
# mass of arithmetic in an effort to crank out a numerical approximation to the 
# solution. The syntax is not obvious, and it takes some getting used to. For now, 
# pretend that you've got these mechanics down. You'd key in this, perhaps:
>  with(DEtools):with(plots):\
de:=D(y)(x) = x^2+(y(x))^2;

                                            2       2
                           de := D(y)(x) = x  + y(x)
> p:= dsolve({de, y(0)=1}, y(x),type=numeric):\
odeplot(p,[x,y(x)],-1..1 );
> 
# Whoa! This is strange. It almost looks like a vertical asymptote, even though there is 
# no hint of division by zero in the problem. In fact, there is a vertical asymptote, and 
# the true solution has shot off to infinity even before x reaches 1.  
# One of the main theorems about differential equations states that under certain 
# reasonable restrictions on the differential equation and its initial conditions (such as 
# the one here that y(0)=1),  a solution is guaranteed to exist, at least for some interval 
# on either side of the x value for the initial conditions. This theorem does not promise 
# that the solution will behave well all the way from one end of the real number line to 
# the other. 
# All too often, something blows up or goes haywire at some point. Numerical 
# procedures, unfortunately, proceed with antlike determination and yield numbers 
# even when the solution has ceased to exist.
# 
# There are other potentially helpful commands in the Maple DE toolkit. For instance, 
# the command DEplot1 shows what is called a direction field for the differential 
# equation, together with some numerically calculated solutions. Here, the symbol 
# string "de" still refers to the differential equation we looked at earlier.
> de;
> DEplot1(de,y(x),x=-1..1,\
{[0,-1],[0,-3/4],[0,-1/2],[0,0],[0,1/2],[0,3/4],[0,1]},\
y=-2..2);
> 
# But look what a confusing and useless picture we get with the seemingly equally 
# good command string  
> DEplot1(de,y(x),x=-1..1,\
{[0,-1],[0,-3/4],[0,-1/2],[0,0],[0,1/2],[0,3/4],[0,1]});
> 
# The difference is that in the first string, we took into account the fact that the 
# solution through [0,1] skitters off to infinity before x reaches 1, and took the 
# precaution of setting limits -2 and 2 to the y-range.  Sometimes Maple will seem 
# confusing and useless. There is always the chance that the confusion stems more 
# from conceptual difficulties with the mathematics itself, than from technical 
# difficulties with syntax and the like. Step back, stare into space, and let your mind 
# roam a bit. 
>