This Sage quickstart tutorial was developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution-ShareAlike 3.0 license (CC BY-SA).
Solving differential equations is a combination of exact and numerical methods, and hence a great place to explore with the computer. We have already seen one example of this in the calculus tutorial, which is worth reviewing.
sage: y = function('y',x) sage: de = diff(y,x) + y -2 sage: h = desolve(de, y)
Forgetting about plotting for the moment, notice that there are three things one needs to solve a differential equation symbolically:
Since we did not specify any initial conditions, Sage (from Maxima) puts in a parameter. If we want to put in an initial condition, we use ics (for initial conditions). For example, we set ics=[0,3] to specify that when \(x=0\), \(y=3\).
sage: h = desolve(de, y, ics=[0,3]); h (2*e^x + 1)*e^(-x)
And of course we have already noted that we can plot all this with a slope field.
sage: y = var('y') # Needed so we can plot sage: Plot1=plot_slope_field(2-y,(x,0,3),(y,0,5)) sage: Plot2=plot(h,x,0,3) sage: Plot1+Plot2 Graphics object consisting of 2 graphics primitives
Regarding symbolic functions versus symbolic variables:
There are many other differential equation facilities in Sage. We can’t cover all the variants of this in a quickstart, but the documentation is good for symbolic solvers.
For instance, Maxima can do systems, as well as use Laplace transforms, and we include versions of these wrapped for ease of use.
In all differential equation solving routines, it is important to pay attention to the syntax! In the following example, we have placed the differential equation in the body of the command, and had to specify that f was the dependent variable (dvar), as well as give initial conditions \(f(0)=1\) and \(f'(0)=2\), which gives the last list in the example.
sage: f=function('f',x) sage: desolve_laplace(diff(f,x,2) == 2*diff(f,x)-f, dvar = f, ics = [0,1,2]) x*e^x + e^x
sage: g(x)=x*e^x+e^x sage: derivative(g,x,2)-2*derivative(g,x)+g x |--> 0
There are also numerical methods.
For instance, one of the options above was desolve_rk4. This is a fourth-order Runge-Kutta method, and returns appropriate (numerical) output. Here, we must give the dependent variable and initial conditions.
sage: y = function('y',x) sage: de = diff(y,x) + y -2 sage: h = desolve_rk4(de, y, step=.05, ics=[0,3])
It can be fun to compare this with the original, symbolic solution. We use the points command from the advanced plotting tutorial.
sage: h1 = desolve(de, y, ics=[0,3]) sage: plot(h1,(x,0,5),color='red')+points(h) Graphics object consisting of 2 graphics primitives
The primary use of numerical routines from here is pedagogical in nature.
For more advanced numerical routines, we primarily use the GNU scientific library. Using this is a little more sophisticated, but gives a wealth of options.
We can even do power series solutions. In order to do this, we must first define a special power series ring , including the precision.
sage: R.<t> = PowerSeriesRing(QQ, default_prec=10) sage: a = -1 + 0*t sage: b = 2 + 0*t sage: h=a.solve_linear_de(b=b,f0=3,prec=10)
This power series solution is pretty good for a while!
sage: h = h.polynomial() sage: plot(h,-2,5)+plot(2+e^-x,(x,-2,5),color='red',linestyle=':',thickness=3) Graphics object consisting of 2 graphics primitives
This was just an introduction; there are a lot of resources for differential equations using Sage elsewhere, including a book by David Joyner, who wrote much of the original code wrapping Maxima for Sage to do just this.