Boyce/DiPrima 10th ed, Ch 7.1: Introduction to Systems of
First Order Linear Equations
Elementary Differential Equations and Boundary Value Problems, 10 th edition, by William E. Boyce and Richard C. DiPrima, ©2013 by John Wiley & Sons, Inc.
• A system of simultaneous first order ordinary differential
equations has the general form
x1  F1 (t , x1 , x2 ,  xn )
x2  F2 (t , x1 , x2 ,  xn )

xn  Fn (t , x1 , x2 ,  xn )
where each xk is a function of t. If each Fk is a linear
function of x1, x2, …, xn, then the system of equations is said
to be linear, otherwise it is nonlinear.
• Systems of higher order differential equations can similarly
be defined.
Example 1
• The motion of a certain spring-mass system from Section 3.7
was described by the differential equation
u(t )  0.125 u(t )  u (t )  0
• This second order equation can be converted into a system of
first order equations by letting x1 = u and x2 = u'. Thus
x1  x2
x2  0.125 x2  x1  0
or
x1  x2
x2   x1  0.125 x2
Nth Order ODEs and Linear 1st Order
Systems
• The method illustrated in the previous example can be used
to transform an arbitrary nth order equation

y ( n )  F t , y, y, y,, y ( n1)

into a system of n first order equations, first by defining
x1  y, x2  y, x3  y, , xn  y ( n 1)
Then
x1  x2
x2  x3

xn 1  xn
xn  F (t , x1 , x2 ,  xn )
Solutions of First Order Systems
• A system of simultaneous first order ordinary differential
equations has the general form
x1  F1 (t , x1 , x2 , xn )

xn  Fn (t , x1 , x2 , xn ).
It has a solution on I:  < t <  if there exists n functions
x1  1 (t ), x2  2 (t ), , xn  n (t )
that are differentiable on I and satisfy the system of
equations at all points t in I.
• Initial conditions may also be prescribed to give an IVP:
x1 (t0 )  x10 , x2 (t0 )  x20 ,, xn (t0 )  xn0
Theorem 7.1.1
• Suppose F1,…, Fn and F1/x1,…, F1/xn,…, Fn/ x1,…,
Fn/xn, are continuous in the region R of t x1 x2…xn-space
defined by  < t < , 1 < x1 < 1, …, n < xn < n, and let the
point
t , x , x ,, x 
0
0
1
0
2
0
n
be contained in R. Then in some interval (t0 - h, t0 + h) there
exists a unique solution
x1  1 (t ), x2  2 (t ), , xn  n (t )
that satisfies the IVP.
x1  F1 (t , x1 , x2 ,  xn )
x2  F2 (t , x1 , x2 ,  xn )

xn  Fn (t , x1 , x2 ,  xn )
Linear Systems
• If each Fk is a linear function of x1, x2, …, xn, then the
system of equations has the general form
x1  p11 (t ) x1  p12 (t ) x2    p1n (t ) xn  g1 (t )
x2  p21 (t ) x1  p22 (t ) x2    p2 n (t ) xn  g 2 (t )

xn  pn1 (t ) x1  pn 2 (t ) x2    pnn (t ) xn  g n (t )
• If each of the gk(t) is zero on I, then the system is
homogeneous, otherwise it is nonhomogeneous.
Theorem 7.1.2
• Suppose p11, p12,…, pnn, g1,…, gn are continuous on an
interval I:  < t <  with t0 in I, and let
x10 , x20 ,, xn0
prescribe the initial conditions. Then there exists a unique
solution
x1  1 (t ), x2  2 (t ), , xn  n (t )
that satisfies the IVP, and exists throughout I.
x1  p11 (t ) x1  p12 (t ) x2    p1n (t ) xn  g1 (t )
x2  p21 (t ) x1  p22 (t ) x2    p2 n (t ) xn  g 2 (t )

xn  pn1 (t ) x1  pn 2 (t ) x2    pnn (t ) xn  g n (t )