Transcript Math 260 - Essex County College

Ch 7.1: Introduction to Systems of First
Order Linear Equations
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 spring-mass system from Section 3.8 was
described by the equation
u(t )  16u(t )  192u(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  16 x2  192x1  0
or
x1  x2
x2  16 x2  192x1
Nth Order ODEs and Linear 1st Order Systems
The method illustrated in 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
Example 2
The equation
y  y  0, 0  t  2
can be written as system of first order equations by letting
x1 = y and x2 = y'. Thus
x1  x2
x2   x1
A solution to this system is
x1  sin(t ), x2  cos(t ), 0  t  2
which is a parametric description
for the unit circle.
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
t0 , x10 , x20 ,, xn0 
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 )