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Serie solution for lineal ODE with non-constant coefficients.

Developments around the origin, if it is a non-singular point. The validity is very limited to a small interval, but it is often useful to have a simple expression for the function near the origin.

Coeffs(P,x) returns the coefficients of the polynom P(x) P# x# M# Coeffs f#(x# num2str): P# num2str concat str2num t# 0 : n# 1 : c# 0 f# 11 mat : 13 mat k# 1 M# range c# k# 1 + el 1 k#! / t# f# t# k# diff * : c# k# 1 + el num2str 0 ≠ n# k# 1 + : if 21 line for c# 1 n# range el 41 line :

Inefficient code, but short.

Coeffs(P,x) returns the coefficients of the polynom P(x) P# x# Coeffs P# x# 20 Coeffs 11 line :

Coeffs(P,x) returns the coefficients of the polynom P(x) P# Coeffs P# P# Unknowns 1 el Coeffs 11 line :

Series develop of a lineal ODE around the origin. deq# yt# IC# n# LODES A# 0 : Y# 0 : k# 1 n# range : y'# yt# : ys# yt# num2str : 15 mat ts# ys# (strsplit 2 el) strsplit 1 el : t# ts# str2num : 12 mat j# 1 n# range y'# y'# num2str ('(strrep : δ# y'# :diff(ys#, ts#, j# num2str) concat str2num : 21 line for k# A# Y# Y# k# el t# k# 1 -^k# 1 - (! / IC# k# el k# IC# length ≤ A# k# el cases * : (sum : A# k# el a#.@ @ k# num2str strrep str2num : σ# equrep(deq#,@1,@2) @2 k# A# Y# num2str strrep : C# σ# @1 ys# strrep str2num t# n# Coeffs : A# k# el a#.@:el(a#,@) @ k# num2str strrep str2num : a# eqa# C# : k# n# 1 matrix eval eqa# al_nleqsolve Y# 101 line :

Example

x u Clear 1

de 2 x u'' * x x u' * + x u + :

IC 11 - 12 mat :

x u.s de x u IC 8 LODES : RK de 0 ≡ 0 u 1 ≡ 0 u' 1 - ≡ 31 sys x u 4200 Rkadapt : 21 line

x u.s x u 2 1 sys

x u.s Coeffs transpose 11 - 0.25 - 0.17 0.03 0.02 - 0018 mat

Example

x u Clear 1

de x cos x u'' * x 1 + (x u' * + 41 x + ln * - :

IC 02 - 12 mat :

x u.s de x u IC 5 LODES : RK de 0 ≡ 0 u 0 ≡ 0 u' 2 - ≡ 31 sys x u 2200 Adams : 21 line

x u.s x u 2 1 sys

x u.s Coeffs transpose 02 - 0.95 0.59 0.3 - 15 mat

Example

de t 1 + (t x'''' * t cos t x'' * + 2 - :

IC 001014 mat :

t x.s de t x IC 7 LODES : RK de 0 ≡ 0 x 0 ≡ 0 x' 0 ≡ 0 x'' 1 ≡ 0 x''' 0 ≡ 51 sys t x 2200 Adams : 21 line

x x.s x x 2 1 sys

t x.s Coeffs transpose 00 0.5 0 0.04 0.01 - 017 mat

Example

Initial point is not the origin:

xo 1 - :

x u Clear 1

de x u'' x x u' * + x 2^x cos * x u * + :

IC 02 - 12 mat :

Change of variable

t x xo - ≡

x u x 2 diff t u t 2 diff ≡

x u x diff t u t diff ≡

and

de_t t u'' t xo + (t u' * + t xo + (2^t xo + cos * t u * + :

t u_t de_t t u IC 6 LODES : x u.s x xo - u_t : RK de 0 ≡ xo u 0 ≡ xo u' 2 - ≡ 31 sys x u 1200 Adams : 31 line

x u.s x u 2 1 sys

x u.s Coeffs transpose 2.66 - 3.17 - 0.51 - 0.26 - 0.36 - 0.11 - 16 mat

Alvaro

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