# (Beginner) Intuition to solve a functional equation and steps for this particular-

Please can someone please tell me the intuition behind solving a functional equation?
For example, $$f(x)+f(y)=2f\left(\frac{x+y}2\right)f'\left(\frac{x-y}2\right)$$ Now at the first look, it seems like $f(x)=\sin x$, but how to solve it rigorously?

P.S: As written in the title, I am a beginner and I have no experience at solving a functional equation. I know little bit about series expansions and I know calculus upto 12th grade. (Derivatives, Integrals (Impropers-a bit), Limits)

• Obviously the identical null function also meets the criterium. At least can we assume $f$ is analytic? Sep 21, 2015 at 11:38
• Sorry, I am a beginner, dunno what is an analytic function. Sep 21, 2015 at 11:39
• I meant if there exist derivatives of $f$. Then you could derive successively, substitute $y$ by $0$ and obtain the repeated derivatives to expand $f$ as a Taylor series around $0$. But I do not think there are systematic "steps" that always lead to a solution of this kind of equations. Sep 21, 2015 at 11:43
• Yes, let us assume that the function is differentiable to lets say, $nth$ degree. Where $n\geq0$. Then? Sep 21, 2015 at 13:16
• Okay, I understand, then what would be the steps of solution here.? Sep 21, 2015 at 13:17

$$f(x)+f(y)=2f\left(\frac{x+y}2\right)f'\left(\frac{x-y}2\right)$$

Let's define $t=\frac{x+y}{2}$, $u=\frac{x-y}{2}$

$$f(t+u)+f(t-u)=2f\left(t\right)f'\left(u\right)$$

for $u=0$

$$f(t)+f(t)=2f\left(t\right)f'\left(0\right)$$ $$2f(t)=2f\left(t\right)f'\left(0\right)$$ $$f'\left(0\right)=1$$

To derivate both side for $u$ $$f'(t+u)-f'(t-u)=2f\left(t\right)f''\left(u\right)$$

Then for $u=0$ again

$$f'(t)-f'(t)=2f\left(t\right)f''\left(0\right)$$

$$0=2f\left(t\right)f''\left(0\right)$$ $$f''\left(0\right)=0$$

To derivate both side for $u$ again $$f''(t+u)+f''(t-u)=2f\left(t\right)f'''\left(u\right)$$

Then for $u=0$ again

$$2f''(t)=2f\left(t\right)f'''\left(0\right)$$

$$f'''\left(0\right)=c^2$$

if $c=0$ then $$f''(t)=0$$ and $$f(t)=a+bt$$ and we know that $$f'\left(0\right)=1$$

Thus $b=1$ $$f(t)=a+t$$

to put into the function equation to find $a$

$$a+x+a+y=2(a+\frac{x+y}2)$$ $$2a+x+y=2a+x+y$$

thus

$$f(t)=a+t$$ is general solution for $c=0$ where $a∈C$ .

if $c \neq0$ then

$$f''(t)=c^2f\left(t\right)$$

The solution of the equation

$$f(t)=k_1e^{ct}+k_2e^{-ct}$$

we know $$f'\left(0\right)=1$$ $$f''\left(0\right)=0$$ and $$f'''\left(0\right)=c^2$$

To solve all $$f'(t)=k_1ce^{ct}-k_2ce^{-ct}$$ $$f'(0)=c(k_1-k_2)$$ $$c(k_1-k_2)=1$$

$$f''(t)=k_1c^2e^{ct}+k_2c^2e^{-ct}$$ $$f''(0)=c^2(k_1+k_2)$$ $$c^2(k_1+k_2)=0$$ $$k_1=-k_2$$

Thus

$$c(k_1-k_2)=1$$ $$2ck_1=1$$ $$k_1=\frac{1}{2c}$$ $c \neq 0$

Our general solution is for $c \neq 0$

$$f(t)=\frac{1}{2c}(e^{ct}-e^{-ct})$$

To confirm the solution

$$\frac{1}{2c}(e^{cx}-e^{-cx})+\frac{1}{2c}(e^{cy}-e^{-cy})=2\frac{1}{2c}(e^{c\frac{x+y}2}-e^{-c\frac{x+y}2})\frac{1}{2}(e^{c\frac{x-y}2}+e^{-c\frac{x-y}2})$$

$$\frac{1}{2c}(e^{cx}-e^{-cx}+e^{cy}-e^{-cy})=\frac{1}{2c}(e^{c\frac{x+y}2}-e^{-c\frac{x+y}2})(e^{c\frac{x-y}2}+e^{-c\frac{x-y}2})$$

$$\frac{1}{2c}(e^{cx}-e^{-cx}+e^{cy}-e^{-cy})=\frac{1}{2c}(e^{cx}+e^{cy}-e^{-cy}-e^{-cx})$$

and you are right that $c=i$ ,$f(t)=\sin(t)$ is a solution of the function equation but not general solution. it is just particular solution

$$f_i(t)=\frac{1}{2i}(e^{it}-e^{-it})=\frac{1}{2i}(\cos{}t +i\sin{t} -\cos{t}+i \sin{t})=\sin{t}$$

The general formula is :

for $c \neq 0$; $$f(t)=\frac{1}{2c}(e^{ct}-e^{-ct})$$

for $c = 0$; $$f(t)=a+t$$ ;where $a∈C$

and also $f(t)=0$ is a solution of the function equation.

• You're missing the solutions $f(x)=0$ and $f(x)=x$. Sep 21, 2015 at 14:10
• @Chappers Thanks for those points Sep 21, 2015 at 14:36

Taking $x=y$ gives $$2f(x) = 2f(x)f'(0), \tag{1}$$ so either $f(x) \equiv 0$ or $f'(0)=1$. Suppose it is the latter (else we have found $f$). Setting $x=-y$ gives $$f(x)+f(-x) = 2f(0)f'(x). \tag{2}$$ Differentiating the original equation with respect to $x$ gives $$f'(x) = f'\left( \frac{x+y}{2} \right)f'\left( \frac{x-y}{2} \right) + f\left( \frac{x+y}{2} \right)f''\left( \frac{x-y}{2} \right), \tag{3}$$ and setting $y=-x$ gives $$f'(x) = f'(0)f'(x)+f(0)f''(x) = f'(x)+f(0)f''(x).$$ This says that either $f''(x) \equiv 0$ or $f(0)=0$. It is easy to check that the only functions with $f''(x) \equiv 0$ which satisfy the equation are $f(x) =0$ and $f(x)=x$, so now assume $f(0)=0$. (2) then gives that $$f(-x) = -f(x),$$ so we also have $f''(0)=0$. Differentiating (3) with respect to $y$ gives $$0 = f''\left( \frac{x+y}{2} \right)f'\left( \frac{x-y}{2} \right) - f'\left( \frac{x+y}{2} \right)f''\left( \frac{x-y}{2} \right) + f\left( \frac{x+y}{2} \right)f''\left( \frac{x-y}{2} \right) - f\left( \frac{x+y}{2} \right)f'''\left( \frac{x-y}{2} \right) \\ = f''\left( \frac{x+y}{2} \right)f'\left( \frac{x-y}{2} \right) - f\left( \frac{x+y}{2} \right)f'''\left( \frac{x-y}{2} \right).$$ Now, if we take $y=x$, this reduces to $$f'''(0) f(x) = f''(x) f'(0) = f''(x).$$ At this point, we can write down the general solution to this equation: setting $f'''(0)=A^2$ gives $$f(x) = B\sinh{Ax}+C\cosh{Ax},$$ and imposing the boundary conditions $f(0)=0$, $f'(0)=1$ gives general solution $$f(x) = \frac{1}{A}\sinh{Ax}.$$ It is simple to check that this always satisfies the functional equation, and further, you can recover the $f(x)=x$ solution by taking the limit as $A \to 0$.

(Remark: this does include the sine solution you expected, since $$\sin{ax} = \frac{\sinh{iax}}{i},$$ so taking $A=ia$ gives the sine solution.

Therefore we have shown

If $f(x)$ is thrice-differentiable and satisfies the functional equation $$f(x)+f(y) = 2f\left( \frac{x+y}{2} \right)f'\left( \frac{x-y}{2} \right),$$ then $f(x)$ is one of

• $0$
• $x$
• $A^{-1}\sinh{Ax}$, some $A \in \mathbb{C}$.

EDIT: In fact, we only need twice-differentiability, because setting $x=0$,$y=2z$ in (3) gives $$1 = (f'(z))^2-f(z)f''(z),$$ which is a thoroughly rotten-looking DE, but we can solve it by noticing that $$\left(\frac{f'}{f}\right)' = \frac{ff''-f'^2}{f^2},$$ so an equivalent is $$-\frac{1}{f^2} = \left(\frac{f'}{f}\right)'$$ Multiply by $2f'/f$: $$-2\frac{f'}{f^3} = 2\left(\frac{f'}{f}\right)\left(\frac{f'}{f}\right)'$$ Integrating this once, we have $$a^2+\frac{1}{f^2} = \frac{f'^2}{f^2},$$ and rearranging this gives $$1 = \frac{f'^2}{a^2 f^2+1}$$ Taking the positive square root (since $f'(0)=1$) and integrating, $$z-0 = \int_0^{f(z)} \frac{dt}{\sqrt{1+a^2 t^2}}$$ Now, this is easy to integrate by setting $u=a^{-1}\sinh{t}$: then the integral just simplifies to $a^{-1}\int_0^{\arg\sinh{(af(z))}} du =\arg\sinh{(af(z))},$ and hence $f(z)=\frac{1}{a}\sinh{az}$ as before. (And $a=0$ gives the $f(x)=x$ solution.)

• Can you explain how can we take $x=\pm y$? Sep 21, 2015 at 14:05
• $x$ and $y$ are just two variables, and the functional equation holds for each of their values, so it must hold for the subset of values where $x=\pm y$. Sep 21, 2015 at 14:07