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单词 ExampleOfSolvingAFunctionalEquation
释义

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:

f(x+y)f(x-y)-f(x-y)f(x+y)=-2f(y)f(y)
f′′(x+y)f(x-y)+f(x-y)f(x+y)-f′′(x-y)f(x+y)-f(x+y)f(x-y)=0

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

f′′(x+y)/f(x+y)=f′′(x-y)/f(x-y).

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

f′′(u)f(u)=f′′(v)f(v)

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 equation  f′′(t)/f(t)=±k2 or

f′′(t)=±k2f(t),

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”

    (x+y)(x-y)=x2-y2.
  2. 2.

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

    sin(x+y)sin(x-y)=sin2x-sin2y,

    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

    sinh(x+y)sinh(x-y)=sinh2x-sinh2y.

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

Titleexample of solving a functional equation
Canonical nameExampleOfSolvingAFunctionalEquation
Date of creation2013-03-22 15:30:00
Last modified on2013-03-22 15:30:00
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id14
Authorpahio (2872)
Entry typeExample
Classificationmsc 34A30
Classificationmsc 39B05
Related topicChainRule
Related topicAdditionFormula
Related topicSubtractionFormula
Related topicDefinitionsInTrigonometry
Related topicGoniometricFormulae
Related topicDifferenceOfSquares
Related topicAdditionFormulas
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更新时间:2025/5/5 2:49:40