getting Taylor series from differential equation


If a given functionMathworldPlanetmath f satisfies a differential equationMathworldPlanetmath, the Taylor seriesMathworldPlanetmath of f can sometimes be obtained easily.

Let

f(x)=sin(marcsinx),

where m is a non-zero , be an example (cf. (http://planetmath.org/Cf) the cyclometric functions).  We form the derivatives

f(x)=m1-x2cos(marcsinx),
f′′(x)=-m21-x2sin(marcsinx)+mx(1-x2)1-x2cos(marcsinx),

which show that f satisfies the differential equation

(1-x2)f′′-xf+m2f=0.

Differentiating this repeatedly gives the equations

(1-x2)f′′′-3xf′′+(m2-1)f=0,
(1-x2)f(4)-5xf′′′+(m2-4)f′′=0,

and so on.  Using the sum of odd numbers1+3+5++(2n-1)=n2  and induction on n yields the recurrence relation

(1-x2)f(n+2)-(2n+1)xf(n+1)+(m2-n2)f(n)=0.

Plugging in   x=0  yields

f(n+2)(0)=(n2-m2)f(n)(0)(n=0, 1, 2,).

Since  f(0)=m,  we have that

f(2n+1)(0)=m(12-m2)(32-m2)((2n-1)2-m2),

whereas all even derivatives of f vanish at x=0.  (Note that f is an odd functionMathworldPlanetmath.)  Thus, we obtain the Taylor of f:

sin(marcsinx)=m1!x+m(12-m2)3!x3+m(12-m2)(32-m2)5!x5+

By the ratio test, this series converges for  |x|<1.

References

  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset I.  WSOY. Helsinki (1950).
Title getting Taylor series from differential equation
Canonical name GettingTaylorSeriesFromDifferentialEquation
Date of creation 2013-03-22 15:06:07
Last modified on 2013-03-22 15:06:07
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 17
Author Wkbj79 (1863)
Entry type Example
Classification msc 41A58
Related topic ExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries
Related topic SawBladeFunction
Related topic SpecialCasesOfHypergeometricFunction