Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 4
Linear Equations
Recall that a function y is linear in a variable x if it describes a straight
line, e.g., we write something like
y = Ax + B (4.1)
to mean that y is linear in x. If we have an algebraic system that depends
on t, we might allow A and B to be functions of t, e..g, the equation
y = A(t)x + B(t) (4.2)
is also linear in x. For example,Recall that a function is linear in a
y = t 2 x + 3 sin t (4.3)
is linear in x because for any fixed value of t, it has the form
y = Ax + B (4.4)
Thus to determine the linearity of a function in x, the nature of the dependence
on any other variable does not matter. The same definition holds for
differential equations.
By a linear differential equation we mean a differential equation of
a single variable, say y(t), whose derivative depends on itself only
linearly. The nature of the dependence on the time variable does not
matter. From our discussion above, something is linear in x if it is written
as
Ax + B (4.5)
25