Example 3. Solve the equation

$$
2 f(x+y)+f(x-y)=f(x)\left(2 e^{y}+e^{-y}\right)
$$

28
Solution. Performing the following substitutions:

$$
x=0, y=t ; x=t, y=2 t ; x=t, y=-2 t
$$

we obtain the equations:

$$
\begin{aligned}
& 2 f(t)+f(-t)=a\left(2 e^{t}+e^{-t}\right) \\
& 2 f(3 t)+f(-t)=f(t)\left(2 e^{2 t}+e^{-2 t}\right) \\
& 2 f(-t)+f(3 t)=f(t)\left(2 e^{-2 t}+e^{2 t}\right)
\end{aligned}
$$

where $a=f(0)$.

We eliminate $f(-t)$ and $f(3 t)$ from these equations. For this, we subtract the third equation, multiplied by 2, from the sum of the first equation, multiplied by 3, and the second equation.

We obtain the general solution in implicit form:

$$
6 f(t)=3 a\left(2 e^{t}+e^{-t}\right)-3 f(t) \cdot e^{-2 t}
$$

from which

$$
f(t)=a \cdot \frac{2 e^{t}+e^{-t}}{2+e^{-2 t}}=a \cdot e^{t}
$$

This function indeed satisfies the original equation, as can be easily verified by checking:

$$
2 a e^{x+y}+a e^{x-y}=a \cdot e^{x}\left(2 e^{y}+e^{-y}\right)
$$

## EXERCISES

53. Show that the following functional equations

a) $f(x+y)=f(x)+f(y)$

b) $f(x y)=f(x) \cdot f(y)$

c) $f(x+y)=f(x) \cdot f(y)$;

d) $f(x y)=f(x)+f(y)$

e) $f(x+y)+f(x-y)=2 f(x) \cdot f(y)$;

f) $f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)}$

are satisfied respectively by the functions

$$
a x, x^{a}, a^{x}, \log _{a} x, \cos x, \operatorname{tg} x
$$

Consider some properties of these functions based on the corresponding functional equation (e.g., evenness, oddness, etc.).

54. Solve the following functional equations:

a) $f\left(\frac{x}{x-1}\right)=2 f\left(\frac{x-1}{x}\right)$;

b) $f(x)+f\left(\frac{4}{2-x}\right)=x$;

c) $a f(x)+f\left(\frac{1}{x}\right)=a x$;
d) $f(n)=f(n-1)+a^{n}, \quad f(1)=1, \quad n \in \mathbf{N}, a \in \mathbf{R}$;

e) $n^{f(n)-1}=(n-1)^{\{(n-1)}, n \in \mathbf{N}$;
f) $f(x+y)+f(x-y)=2 f(x) \cdot \cos y$.

55. Prove that if for $x \in \mathbf{R}$ and a constant $m \in \mathbf{R}$ the equality $f(x+m)=\frac{1+f(m)}{1-f(m)}$ holds, then $f(x)$ is a periodic function. Find an example of such a function.

56*. Solve the following functional equations:
a) $f^{2}(x)=f(2 x)$

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

c) $f(x+y)+2 f(x-y)=3 f(x)-y$;

d) $f(x+y)=f(x)+y$

e) $f(x+y)-2 f(x-y)+f(x)-2 f(y)=y-2$.

## § 5. "Non-standard" equations and systems of equations

There is no general theory for solving equations and systems of equations of the considered type. In the process of solving them, such properties of the functions involved in the equation are used as boundedness (from above or below), monotonicity, evenness or oddness, etc.

Consider examples.