Introductory Differential Equations using Sage - William Stein

1.4. FIRST ORDER ODES - SEPARABLE AND LINEAR CASES 29
2
1.5
1
0.5
1 2 3 4
Figure 1.8: Plot of y ′ = y(y − 1)(y − 2), y(0) = y 0 , for 0 < t < 4, and various values of y 0 .
There are a variety of very specialized substitution methods which we will not present in
this text. However one class of differential equations is common enough to warrant coverage
here: homogeneous ODEs. Unfortunately, for historical reasons the word homogeneous has
come to mean two different things. In the present context, we define a homogeneous firstorder
ODE to be one which can be written as y ′ = f(y/x). For example, the following ODE
is homogeneous in this sense:
dy
dx = x/y.
A first-order homogeneous ODE can be simplified by using the substitution v = y/x, or
equivalently y = vx. Differentiating that relationship gives us v + xv ′ = y ′ .
Example 1.4.5. We will solve the ODE y ′ = y/x + 2 with the substitution v = y/x.
As noted above, with this substituion y ′ = v +xv ′ so the original ODE becomes v +xv ′ =
v + 2. This simplifies to v ′ = 2/x which can be directly integrated to get v = 2log(x) + C
(in more difficult examples we would get a separable ODE after the substitution). Using
the substitution once again we obtain y/x = 2log(x) + C so the general solution is y =
2xlog(x) + Cx.
1.4.4 Linear 1st order ODEs
The bottom line is that we want to solve any problem of the form
x ′ + p(t)x = q(t), (1.5)
where p(t) and q(t) are given functions (which, let’s assume, aren’t “too horrible”). Every
first order linear ODE can be written in this form. Examples of DEs which have this form:
Falling Body problems, Newton’s Law of Cooling problems, Mixing problems, certain simple
Circuit problems, and so on.