Lecture Notes in Differential Equations - Bruce E. Shapiro

26 LESSON 4. LINEAR EQUATIONS
where A and B do not depend on x. Hence something is linear in y if it
can be written as
Ay + B (4.6)
if A and B do not depend on y. Thus for a differential equation to be linear
in y it must have the form
dy
dt
= Ay + B (4.7)
where neither A nor B depends on y. However, both A and B are allowed
to depend on t. To emphasize this we write the terms A and B as A(t)
and B(t), so that the linear equation becomes
dy
dt
= A(t)y + B(t) (4.8)
For convenience of finding solutions (its not clear now why this is convenient,
but trust me) we bring the term in A(t)y to the left hand side of the
equation:
dy
− A(t)y = B(t) (4.9)
dt
To be consistent with most textbooks on differential equations we will relabel
A(t) = −p(t) and B(t) = q(t), for some function p(t) and q(t), and
this gives us
dy
+ p(t)y = q(t) (4.10)
dt
which we will refer to as the standard form of the linear ODE. Sometimes
for convenience we will omit the t (but the dependence will be implied) on
the p and q and write the derivative with a prime, as
y ′ + py = q (4.11)
There is a general technique that will always work to solve a first order
linear ODE. We will derive the method constructively in the following paragraph
and then give several examples of its use. The idea is look for some
function µ(t) that we can multiply both sides of the equation:
µ × (y ′ + py) = µ × q (4.12)
or
µy ′ + µpy = µq (4.13)