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.