Rellich's theorem

In mathematical analysis and differential calculus, the Rellich theorem is a theorem on entire solutions of a differential equation , proved in 1940 by Franz Rellich .

Wording

Let in a differential equation

{\dot {x}}=f(x,t)

{\ displaystyle {\ dot {x}} = f (x, t)}

the right-hand side is everywhere convergent power series on $x, t$ ${\ displaystyle x, t}$ ( whole function ). If there are two solutions $x=u(t)$ ${\ displaystyle x = u (t)}$ and $x=v(t)$ ${\ displaystyle x = v (t)}$ which are entire functions $t$ ${\ displaystyle t}$ then any other whole solution $x=w(t)$ ${\ displaystyle x = w (t)}$ has the form

w(t)=u(t)+(v(t)-u(t))c

{\ displaystyle w (t) = u (t) + (v (t) -u (t)) c}

with a properly selected constant $c$ ${\ displaystyle c}$ . If a $f(x,t)$ ${\ displaystyle f (x, t)}$ not a linear function $x$ ${\ displaystyle x}$ , then there is no more than a countable number of constants $c_{n}$ ${\ displaystyle c_ {n}}$ for which the expression

u(t)+(v(t)-u(t))c_{n}

{\ displaystyle u (t) + (v (t) -u (t)) c_ {n}}

is a solution and set $c_{n}$ ${\ displaystyle c_ {n}}$ cannot have an ending limit point .

The last statement can be reversed: there always exists a nonlinear differential equation with an integer right-hand side having an infinite series of integer solutions $u(t)+(v(t)-u(t))c_{n}$ ${\ displaystyle u (t) + (v (t) -u (t)) c_ {n}}$ for any given $u(t),v(t)$ ${\ displaystyle u (t), v (t)}$ not equal to each other at any value $t$ ${\ displaystyle t}$ , and any set of numbers $c_{n}$ ${\ displaystyle c_ {n}}$ (with a limit point except at infinity).

Consequences

A consequence of the Rellich theorem is that the general solution $x=x(t,C)$ ${\ displaystyle x = x (t, C)}$ nonlinear equation ${\dot {x}}=f(x,t)$ ${\ displaystyle {\ dot {x}} = f (x, t)}$ with the whole right-hand side it cannot be an entire function of t , while any linear differential equation with integer coefficients always has an integer common solution.

Links

Rellich, Fr. Ueber die ganzen Loesungen einer gewoehnlichen Differentialgleichung erster Ordnung (German) // Math. Ann. . - 1940 .-- T. 117 . - S. 587-589 .
Wittich G. Chapter V. Applications to ordinary differential equations // The latest research on unique analytic functions . - M .: Fizmatlit, 1960. - p. 114.

Rellich's theorem

Wording

Consequences

Links

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