Maximum Calculus with Maxima

We looked at Maxima in the February 2011 issue to do algebra and rearrange some equations. But those aren't the only tricks up Maxima's sleeve. This month, I describe how Maxima can help with differential equations, but I'm going to leave out some of the intermediate results to save some space.

A lot of science involves figuring out how systems change over time and what causes those changes. When you start looking at changes, and especially rates of change, that is essentially calculus. Calculus and rates of change also are linked to slopes of lines on graphs. When you plot data and find an equation that describes it, you can find the slope of the line by taking the derivative of the equation. Let's look at a falling object and see what theory has to say about it.

You should start by looking at how you get a derivative. Let's say you have the equation:

(%i1) f(x):= 2 + x^2;
                                          2
(%o1)                        f(x) := 2 + x

You would find the first derivative by calling the function diff, giving it the equation to differentiate along with the variable to differentiate by. So, you would write:

(%i2) answer:diff(f(x),x);
(%o2)                          2x

Maxima can do differentiation of expressions too. If you have a couple equations, you can derive their ratio with:

(%i3) g(x):= x^(1/2);
(%i4) ratio_diff:diff(g(x)/f(x),x);
                                        3/2
                     1               2 x
(%o4)        ----------------- - -----------
                         2          2    2
             2 sqrt(x) (x + 1)    (x + 1)

This might be a bit messy to work with, so you might want to refactor it to a more concise form:

(%i5) factor(ratio_diff);
                                2
                             3 x - 1
(%o5)               - ------------------
                                  2    2
                      2 sqrt(x) (x + 1)

Maxima also can handle trigonometric functions, but there are lots of identities you can use to help simplify equations with trig functions in them. By default, Maxima does not try to apply these unless you specifically say so, using special functions. As an example, let's say you have the following equation:

(%i6) diff(sin(x)/(1 + cos(x)),x);
                                2
                             sin (x)       cos(x)
(%o6)                   ------------- + ----------
                                    2   cos(x) + 1
                        (cos(x) + 1)
(%i7) factor(%);
                            2         2
                         sin (x) + cos (x) + cos(x)
(%o7)                    --------------------------
                                          2
                              (cos(x) + 1)

That's still not very simple. If you then apply the function trigsimp, you can force Maxima to apply trigonometric simplification rules to the equation and see what you get:

(%i8) trigsimp(%);
                             1
(%o8)                   ----------
                        cos(x) + 1

You should be aware of some important caveats regarding how Maxima treats trig functions. The first is that sin(x)^(-1) is the reciprocal of sine, not arcsine. To get the arcsine, you would use asin(x). The other is another trig simplification function, trigreduce. This function is used to reduce the powers of trig functions by using the multiple angle formulas. For example:

(%i9) trigsimp(cos(x)^2 + 2*sin(x)^2);
                          2
(%o9)                  sin (x) + 1
(%i10) trigreduce(cos(x)^2 + 2*sin(x)^2);
                   cos(2 x) + 1      1   cos(2 x)
(%o10)             ------------ + 2 (- - --------)
                         2           2       2

That may not look simpler than what you would get from trigsimp, but it is an easier form of the equation to use with other functions, like integration.

Maxima can apply the chain rule when doing a derivative. Say you have the equation:

(%i11) f(x):= x^3);
                                3
(%o11)                 f(x) := x
(%i12) depends(x,u)$
(%i13) diff(f(x),u);
                         2 dx
(%o13)                3 x  --
                           du

The line at %i12 uses a new function, depends. This is a way of telling Maxima that x is a function of u, without explicitly defining a function describing this relationship. If you decide later that you want to define an actual equation for this relation, you can use:

(%i14) remove([x,u],dependency);
(%o14)                     done
(%i15) x:sin(u);
(%o15)                   sin(u)
(%i16) diff(f(x),u);
                               2
(%o16)             3 cos(u) sin (u)

Along the same lines, Maxima can handle implicit differentiation. Say you have the equation x^2 + y^2 = 25, and you want to find dy/dx. You need to use the depends function I just mentioned to handle this:

(%i17) eqn := x^2 + y^2 = 25;
                    2   2
(%o17)             y + x = 25
(%i18) depends(y,x);
(%o18)              [y(x)]
(%i19) deriv_of_eqn:diff(eqn,x);
                        dy
(%o19)              2 y -- + 2 x = 0
                        dx
(%i20) solve(deriv_of_eqn,'diff(y,x));
                       dy     x
(%o20)                [-- = - -]
                       dx     y

The other side of calculus is integration. The basic function to do that in Maxima is called integrate. This function can do both definite and indefinite integrals. Indefinite integrals are the symbolic form of integration you likely learned in school. For example:

(%i21) integrate(x^2,x);
                           3
                          x
(%o21)                    -
                          3

A definite integral actually is evaluated over an interval. This form of an integral can be visualized as the area under the curve defined by the equation you are integrating. To do definite integrals, simply add two arguments giving the start and end points of the interval:

(%i22) integrate(x^2,x,0,1);
                          1
(%o22)                    -
                          3

Putting all these techniques together, you can solve a differential equation for a given variable—for example, solve dy/dx = f(x) for y. You can do this by doing all the required algebra and calculus, but you don't really need to. Maxima has the very powerful function, ode2, which can do it in one step. Start with your equation: Garrick, shrink below.

(%i23) eq: 'diff(y,x) = sqrt(1/x^2 - 1/x^3);
                      dy        1    1
(%o23)                -- = sqrt(-- - --)
                      dx         2    3
                                x    x
(%i24) ode2(eq,y,x);
                                                     2
                          2                  2 sqrt(x - x)
(%o24)    y = log(2 sqrt(x - x) + 2 x - 1) - ------------- + %c
                                                   x

This one function call does the integration and the solve steps and gives you a final answer to the differential equation.

Let's say you're doing an experiment dropping a coin and timing how long it takes to fall. How do you know whether the times you are measuring actually make sense? Let's start with the most basic law: force = mass * acceleration.

The mass of the coin is a constant, so ignore that for now. The force is the force due to gravity, pulling the coin down to the ground, and the acceleration describes the coin's motion due to this force. The force due to gravity is a constant, at least here on Earth, and it depends linearly on the mass, so you can define the force as:

(%i1) force: mass * g;
(%o1)                g mass

The acceleration also is a constant, because both the mass and the force are constants. Acceleration is simply the rate of change of the velocity, and the velocity is the rate of change of the position, so you can set that up as:

(%i2) depends(y,t);
(%o2)                [y(t)]
(%i3) acceleration: 'diff('diff(y,t),t);
                       2
                      d y
(%o3)                 ---
                        2
                      dt

Putting it all together, you get:

(%i4) eq_of_motion: force = mass * acceleration;
                                        2
                                       d y
(%o4)                    g mass = mass ---
                                         2
                                       dt
(%i5) solve(eq_of_motion, y);
                            2
                           d y
(%o5)                     [--- = g]
                             2
                           dt

You can see right away that how fast an object falls doesn't depend on the mass at all. Galileo was right! The next step is to do some integrating and see what you end up with:

(%i6) integrate(%,t);
                 dy
(%o6)           [-- = g t + %c1]
                 dt

At this step, you would be able to find out the velocity (dy/dt) at time t. The additional term %c1 is a constant of integration. In this case, you can see that it represents the initial velocity of your penny. One more round of integration gives this:

(%i7) integrate(%,t);
                   /            2
                   [ dy      g t
(%o7)            [I  -- dt = ---- + %c1 t + %c2]
                   ] dt       2
                   /

Now you can find the position, y, of your coin at any time, t. Again, a new constant of integration is introduced, %c2. In this case, you can see that this represents the starting height of your coin. But that's not what you were measuring. You were measuring how long it took the coin to drop a given distance. So you need to do a bit of rearranging. Because you are dropping your coin, you know that the start velocity is 0 (that is, %c1=0). You can rewrite things a little to make it a bit clearer: Garrick, shrink below.

(%i8) eqn: y = (g * t^2)/2 + y0;
                                 2
                              g t
(%o8)                y = y0 + ----
                               2
(%i9) solve(eqn,t);
                             y   y0                    y   y0
(%o9)    [t = - sqrt(2) sqrt(- - --), t = sqrt(2) sqrt(- - --)]
                             g   g                     g   g

There you go. You now have an equation for the time, given a height that your coin is dropping. With this theoretical relation under your belt, you can check to see whether gravity is working correctly in your local lab. If not, you should contact the Nobel committee straightaway.

This only scratches the surface of Maxima's capabilities in dealing with calculus and differential equations, but hopefully, this article gives you a starting point. Happy integrating.