Chapter 7
Systems of First-Order Differential
Equations
In this chapter we shall attempt to answer the following questions:
"What is a system of first-order differential equations?"
"What is a linear system of first-order differential equations?"
"What is a solution of a system of first-order differential equations?"
"What is an initial value problem for a system of first-order differential
equations?"
"Under what conditions does a solution to a system initial value problem
exist and under what conditions is the solution unique?"
"Where-that is, on what interval or what region- does the solution to a
system initial value problem exist and where is the solution unique?"
"How can an n-th order differential equation be rewritten as an equivalent
system of first-order differential equations?"DEFINITION System of n First-Order Differential Equations
A system of n first-order differential equations has the form(1)Y~ = fn(x,y1,Y2, ... ,yn)
where each dependent variable Yi is a real-valued function of the independentvariable x and each fi is a real-valued function of x, Yi, Y2, ... , Yn·
Systems of differential equations of this type arise naturally when there is one
independent variable, such as time, and several dependent variables, such as
position and velocity in multidimensional space.
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