Coupled ODE Tutorial
1S2
JF Natural Sciences
1. Confirm by substitution that, for any values of the constants a and b, the
expressions
x = a e2t − b e−3t
y = a e2t + 4b e−3t
solve the following linear system of first order linear differential equations
dx
= x+ y
dt
dy
= 4x − 2y .
dt
2. Write the following pair of linear differential equations in matrix form
dx
= −4x + y
dt
dy
=
4x − 4y .
dt
by indicating the matrix elements of the matrix A appearing in
d
dt
"
x
y
#
= A
"
x
y
#
.
3. Find the eigenvalues λ and λ0 of the matrix A that characterise the odes.
4. Obtain eigenvectors V and V 0 of the matrix A corresponding to λ and λ0 .
5. Show by substitution that the following expression solves the above matrix
form of the equations, where c and c0 are arbitrary constants.
"
x
y
#
0
= c V eλ t + c0 V 0 eλ t .
6. Determine the solution of the coupled system of odes which, at time t = 0,
satisfies the initial conditions x(0) = 7 and y(0) = 10.
Background Material on Coupled Differential Equations
• A coupled system of ordinary differential equations (ODEs) involves a relation between two dependent variables x and y and their derivatives with
respect to an independent variable t.
• The dependent variables x and y often represent the population levels of
two species while their first order derivatives often refer to individual specie
growth rates with variable t indicating the time.
• A general first order system would relate
t,
x,
y,
dx
,
dt
dy
.
dt
• Example 4 of Section 8.10 on page 546 of Contemporary Linear Algebra,
John Wiley, by Howard Anton and Robert Busby [Hamilton Library 512.5
P3] provides a solution of the differential equations appearing in the first
problem overleaf, albeit with a different notation where x and y are denoted
by y1 and y2 , respectively.
• An ‘initial value problem’ refers to an ODE system where the values of x
and y, the specie populations for example, are given at a particular value of
time t, often zero. The constants appearing in the solution of the system
may be determined from such initial values of x and y.
• One way of solving the system of coupled linear first order ordinary differential equations appearing in Question 1 overleaf is provided on page 422 in
section 9.1 of the 8th edition of Anton and Rorres. The method of Anton
and Busby may be simpler.
Dr. Buttimore
1S2
School of Mathematics