deTaylor

[fhub]

Here's a small CAS function deTaylor for the HP-Prime (similiar to deSolve) which I've originally written for Xcas.
It approximately solves 1st- and 2nd-order differential equations (with initial conditions) as n-th degree Taylor polynomial.

1st order y'=f(x,y) with y(x0)=y0:
deTaylor(f,[x,y],[x0,y0],n)

2nd order y''=f(x,y,y') with y(x0)=y0 and y'(x0)=y0':
deTaylor(f,[x,y,z],[x0,y0,z0],n)
(I'm using z instead of y', because y' can't be used as input)

Here's the function definition:
(you can directly copy&paste it into the Prime-emulator commandline in CAS-mode)

Code:


deTaylor(f,v,c,n):=
BEGIN
  local k,yk,sol;
  yk:=v(2); sol:=c(2);
  IF dim(v)<2 or dim(v)>3 or dim(v)<>dim(c) THEN
    RETURN("Argument error!");
  END;
  FOR k FROM 1 TO n STEP 1 DO
    IF dim(v)=2 THEN
      yk:=diff(yk,v(1))+f*diff(yk,v(2));
      sol+=subst(yk,[v(1)=c(1),v(2)=c(2)])/k!*(v(1)-c(1))^k;
    ELSE
      yk:=diff(yk,v(1))+v(3)*diff(yk,v(2))+f*diff(yk,v(3));
      sol+=subst(yk,[v(1)=c(1),v(2)=c(2),v(3)=c(3)])/k!*(v(1)-c(1))^k;
    END;
  END;
  expand(simplify(sol));
END

Example 1: y'=x*y^2+1 with y(0)=1 (6th-degree approximation):
deTaylor(x*y^2+1,[x,y],[0,1],6)

Example 2: y''=x*y*y' with y(1)=2 and y'(1)=3 (5th degree):
deTaylor(x*y*z,[x,y,z],[1,2,3],5)

Maybe it's useful for someone,
Franz