Exercise 7, § 1

Functions of a Real Variable, IV, 200

Consider the system of scalar differential equations

\frac{dx_i}{dt} = \sum_{j,k=1}^n c_{ijk} x_j x_k \quad (1 \leq i \leq n)

where the $c_{ijk}$ are constants such that $c_{kji} = -c_{ijk}$ . Show that the integrals of this system are defined for all values of $t$ (note that $\sum_{i=1}^n x_i^2$ is constant for every integral $x = (x_i)$ ).

Solution

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