Study Guide: Systems of Differential Equations
1. Systems of Differential Equations
A system of differential equations is simply a collection differential equations involving
the same variables y1 , . . . , yn . In most cases, the number of equations is the same as the
number of variables.
The general solution to a system of differential equations is a set of general formulas for
the variables y1 , . . . , yn that satisfy the system. Usually the general solution will have several
arbitrary constants C1 , . . . , Cn , where the number of constants is the same as the number of
variables.
2. Solving for One Variable First
Sometimes one of the equations involves only one of the variables. In this case, you can solve
the system using the following procedure:
1. First, solve the equation that involves only one variable.
2. Substitute the solution into the other equation(s), and then solve those as well.
For example, in the system
y10 = 3y1 ,
y20 = y1 y22
solving the first equation for y1 gives y1 = C1 e3x . Plugging this into the second equation gives
y20 = C1 e3x y22 , which can be solved using separation of variables.
3. Homogeneous Linear Systems
A homogeneous linear system in two variables is a system of the form
0
y10 = ay1 + by2
y1
a b y1
.
or equivalently
=
y20
c d y2
y20 = cy1 + dy2
Homogeneous linear systems involving three or more variables are be defined similarly.
The solutions to such a system are
y1
C1 eλ1 x
= S
,
y2
C2 eλ2 x
a b
where λ1 , λ2 are the eigenvalues of the matrix
, and S is a 2 × 2 matrix whose columns
c d
are eigenvectors for λ1 and λ2 , respectively. A similar solution works for homogeneous linear
systems involving three or more variables.