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.



where m is a non-zero , be an example (cf. ( the cyclometric functions).  We form the derivatives


which show that f satisfies the differential equation


Differentiating this repeatedly gives the equations


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


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


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


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


  • 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
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