example of solving a functional equation

Let’s determine all twice differentiableMathworldPlanetmathPlanetmath real functions f which satisfy the functional equation

f(x+y)f(x-y)=[f(x)]2-[f(y)]2 (1)

for all real values of x and y.

Substituting first  y=0 in (1) we see that  f(x)2=f(x)2-f(0)2  or  f(0)=0.  The substitution  x=0  gives  f(y)f(-y)=-f(y)2,  whence  f(-y)=-f(y).  So f is an odd functionMathworldPlanetmath.

We differentiate both sides of (1) with respect to y and the result with respect to x:


The result is simplified to  f′′(x+y)f(x-y)=f′′(x-y)f(x+y),  i.e.


Denoting  x+y:=u,  x-y:=v  we obtain the equation


for all real values of u and v.  This is not possible unless the proportion f′′(u)f(u) has a on u.  Thus the homogeneous linear differential equationf′′(t)/f(t)=±k2 or


with k some , is valid.

There are three cases:

  1. 1.

    k=0.  Now  f′′(t)0  and consequently  f(t)Ct.  If one especially C equal to 1, the solution is the identity function (http://planetmath.org/IdentityMap)  f:tt.  This yields from (1) the well-known “memory formula”

  2. 2.

    f′′(t)=-k2f(t)  with  k0.  According to the oddness one obtains for the general solution the sine functionf:tCsinkt.  The special case  C=k=1  means in (1) the


    which is easy to verify by using the addition and subtraction formulae (http://planetmath.org/AdditionFormula) of sine.

  3. 3.

    f′′(t)=k2f(t)  with  k0.  According to the oddness we obtain for the general solution the hyperbolic sineMathworldPlanetmath (http://planetmath.org/HyperbolicFunctions) functionMathworldPlanetmathf:tCsinhkt.  The special case  C=k=1  gives from (1) the


The solution method of (1) is due to andik and perucho.

Title example of solving a functional equation
Canonical name ExampleOfSolvingAFunctionalEquation
Date of creation 2013-03-22 15:30:00
Last modified on 2013-03-22 15:30:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Example
Classification msc 34A30
Classification msc 39B05
Related topic ChainRule
Related topic AdditionFormula
Related topic SubtractionFormula
Related topic DefinitionsInTrigonometry
Related topic GoniometricFormulae
Related topic DifferenceOfSquares
Related topic AdditionFormulas