The above theorem is also named the Picard-Lindelöf theorem and can be generalized to a system of first order ordinary differential equations

Theorem 2 (generalization of Picard's theorem [KF])   Let $E$ be an open subset of $\mathbb{R}^{n+1}$ and a continuous function $f(x,y_1,\ldots,y_n)$ defined as $f=(f_1,\ldots,f_n)\colon E \to \mathbb{R}^n$ . If $(t_0,y_{10},\ldots,y_{n0})\in E$ and $f$ satisfies the Lipschitz condition in the variable $y_1,\ldots,y_n$ in $E$ : $$|f_i(x,y_1,\dots,y_n)-f_i(x,y_1'\ldots,y_n')| \leq M \max_{1\leq j\leq n}| y_j-y_j'|$$ where $M$ is a constant. Then the system of ordinary differential equation defined as

with the initial condition $$y_1(x_0) = y_{10},\ldots, y_n(x_0) = y_{n0}$$ has a unique solution $$y_1(x) ,\ldots, y_n(x)$$ on some interval $|x- x_0| \leq\delta$ .

see also:

• the entry existence and uniqueness of solution of ordinary differential equations

References

KF

Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.