4 First-order systems Remark 4.1. Every ODE and every system of ODEs can be reduced to the form y 0 (t) = f (t, y) for some vector y and a vector-valued function f . Consider the third-order ODE x000 (t) − 2x00 (t) + tx(t)2 = 0, for instance. Introducing a vector whose components are x, x0 and x00 , we find that x y1 x0 y2 = y3 = f (t, y). x00 y = x0 = y2 =⇒ y 0 = 00 00 2 2 x y3 2x − tx 2y3 − ty1 Definition 4.2 (Norm). Given a matrix A, we define its norm |A| by X |aij |. |A| = i,j Using this definition, one can easily check that |A + B| ≤ |A| + |B| and |AB| ≤ |A| |B|. Theorem 4.3 (Existence of solutions). Suppose that f , fy1 , . . . , fyn are continuous in an arbitrary set A ⊂ Rn+1 that contains the point (t0 , y0 ). Then the initial value problem y 0 (t) = f (t, y(t)), y(t0 ) = y0 has a unique solution that can be extended for all times, unless either its graph meets the boundary of A or else |y(t)| blows up in finite time.