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Differential Equations

Notation

This page provides examples of how to represent ordinary differential equations (ODEs) in Maxima when writing STACK questions.

Representing ODEs

In a Maxima session we can represent an ODE as

ODE: x^2*'diff(y,x) + 3*y*x = sin(x)/x;

Notice the use of the ' character in front of the diff function to prevent evaluation. Applied to a function call, such as diff, the single quote prevents evaluation of the function call, although the arguments of the function are still evaluated (if evaluation is not otherwise prevented). The result is the noun form of the function call.

Entering DEs

The syntax to enter a derivative in Maxima is diff(y,x,n). Teachers need to use an apostrophe' character in front of the diff function to prevent evaluation in question variables (etc). E.g. to type in you need to use 'diff(y,x,2).

Students' answers always have noun forms added. If a student types in diff(y,x) then this is protected by a special function noundiff(y,x) (etc), and ends up being sent to answer test as 'diff(y,x,1). If a student types in (literally) diff(y,x)+1 = 0 this will end up being sent to answer test as 'diff(y,x,1)+1 = 0.

The answer test AlgEquiv evaluates all nouns. This has a (perhaps) unexpected side-effect that noundiff(y,x) will be equivalent to 0, and noundiff(y(x),x) is not. For this reason we have an alternative answer test AlgEquivNouns which does not evaluate all the nouns. The ATEqualComAss also evaluates its arguments but does not "simplify" them. So, counter-intuitively perhaps, we currently do have ATEqualComAss(diff(x^2,x), 2*x); as true.

Students might expect to enter expressions like , or (especially if you are using derivabbrev:true, see below). The use by Maxima of the apostrophe which affects evaluation also has a side-effect that we can't accept y' as valid student input. Input y_x is an atom. Individual questions could interpret this as 'diff(y,x) but there is no systematic mechanism for interpreting subscripts as derivatives. Input dy/dx is the division of one atom dy by another dx and so will commute with other multiplication and division in the expression as normal. There is no way to protect input dy/dx as . The only input which is interpreted by STACK as a derivative is Maxima's diff function, and students must type this as input.

The expression diff(y(x),x) is not the same as diff(y,x). In Maxima diff(y(x),x) is not evaluated further. Getting students to type diff(y(x),x) and not diff(y,x) will be a challenge. Hence, if you want to condone the difference, it is probably best to evaluate the student's answer in the feedback variables as follows to ensure all occurrences of y become y(x).

ans1:'diff(y(x),x)+1 = 0;
ansyx:subst(y,y(x),ans1);

Trying to substitute y(x) for y will throw an error. Don't use the following, as if the student has used y(x) then it will become y(x)(x)!

ans1:'diff(y,x)+1 = 0;
ansyx:ev(ans1,y=y(x));

Further work is needed to better support partial derivatives (input, display and evaluation).

Displaying ODEs

Maxima has two notations to display ODEs.

If derivabbrev:false then'diff(y,x) is displayed in STACK as . Note this differs from Maxima's normal notation of .

If derivabbrev:true then 'diff(y,x) is displayed in STACK and Maxima as .

  • Extra brackets are sometimes produced around the differential.
  • You must have simp:true otherwise the display routines will not work.

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See also

Maxima reference topics