Computer Algebra Recipes

128 CHAPTER 3. LINEAR ODE MODELS
series solution to order n as long as the percentage error in the previous order
is above 1 percent.
> for n from 1 to 50 while percent[n-1]>1 do
The order of the series is taken to be n. The error in the series is of order x n ,
terms of this order and larger being neglected. If the order is not speci¯ed, the
default order is 6.
> Order:=n:
The di®erential equation is solved for y(x), subject to the initial condition,
with the series option included. The dsolve command uses several methods
when trying to ¯nd a series solution to an ODE or ODE system. When initial
conditions are given, such as in this problem, the series is calculated at the
given point. Otherwise, the series is calculated at the origin. The ¯rst method
used is a Newton iteration, the second involves a direct substitution to generate
a system of equations which must be solved, while the third is the method of
Frobenius discussed in standard ODE texts. Useful references may be found by
entering the topic dsolve,series in Maple's Topic Search.
> sol[n]:=dsolve(fde,icg,y(x),'series');
To remove the \order of" term that appears in the output of the last command
line, the convert( ,polynom) command is applied to the right-hand side of the
nth order solution. The largest term in the series for Yn is x n¡1 .
> Y[n]:=convert(rhs(sol[n]),polynom);
The absolute percentage error at x = X is calculated for each order,
> percent[n]:=abs(evalf(100*(Ynum(X)-eval(Y[n],x=X))/Ynum(X)));
and the do loop ended with a colon to suppress the lengthy output.
> end do:
The loop stops at N = n ¡ 1, in this case order 17, for which the percentage
error (to 3 digits) is 0.696 percent. As you can check by either replacing the
colonwithasemicoloninthedolooportakingN = n¡ 2, the percentage error
in order 16 is 1.31 percent. Order 17 is the ¯rst order for which the percentage
error drops below one percent. The corresponding polynomial representation
of the solution is given by Y17.
> N:=n-1; evalf(percent[N],3)*percent; Y[N]:=Y[N];
N := 17 0:696 percent
Y17 := 1 + 1
15 x3 + 1
70 x4 + 4
1575 x5 + 29
20790 x6 + 43
126126 x7 61
+
1001000 x8
16013
+
1033782750 x9 498307
+
152770117500 x10 7710323
+
14115958857000 x11
2117887
+
21250934424720 x12 96270661
+
5534097506437500 x13 121568353967
+
46021554863534250000 x14
1141494706493
+
2859910909376771250000 x15 2799853112879
+
47283860368362618000000 x16