Теорема Реллиха

В математическом анализе и дифференциальном исчислении теорема Реллиха — теорема о целых решениях дифференциального уравнения, доказанная в 1940 году Францем Реллихом.

Формулировка

Пусть в дифференциальном уравнении

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

правая часть является всюду сходящимся степенным рядом по $x,t$ (целой функцией). Если имеется два решения $x=u(t)$ и $x=v(t)$ , которые являются целыми функциями $t$ , то любое другое целое решение $x=w(t)$ имеет вид

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

при надлежащим образом выбранной константе $c$ . Если $f(x,t)$ не является линейной функцией $x$ , то имеется не более чем счётное число констант $c_{n}$ , при которых выражение

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

является решением и множество $c_{n}$ не может иметь конечной предельной точки.

Последнее утверждение допускает обращение: всегда существует нелинейное дифференциальное уравнение с целой правой частью, имеющее бесконечную серию целых решений $u(t)+(v(t)-u(t))c_{n}$ при любых заданных $u(t),v(t)$ , не равных друг другу ни при каком значении $t$ , и любом наборе чисел $c_{n}$ (имеющих предельную точку разве лишь на бесконечности).

Следствия

Следствием теоремы Реллиха является то, что общее решение $x=x(t,C)$ нелинейного уравнения ${\dot {x}}=f(x,t)$ с целой правой частью не может быть целой функцией от t, в то время как всякое линейное дифференциальное уравнение с целыми коэффициентами всегда имеет целое общее решение.

Ссылки

Rellich, Fr. Ueber die ganzen Loesungen einer gewoehnlichen Differentialgleichung erster Ordnung (нем.) // Math. Ann.. — 1940. — Т. 117. — С. 587—589.
Виттих Г. Глава V. Приложения к обыкновенным дифференциальным уравнениям // Новейшие исследования по однозначным аналитическим функциям. — М.: Физматлит, 1960. — С. 114.

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