Lecture Notes in Differential Equations - Bruce E. Shapiro

2 LESSON 1. BASIC CONCEPTS
Ordinary differential equations are further classified by type and degree.
There are two types of ODE:
• Linear differential equations are those that can be written in a
form such as
a n (t)y (n) + a n−1 (t)y (n−1) + · · · + a 2 (t)y ′′ + a 1 (t)y ′ + a 0 (t) = 0 (1.4)
where each a i (t) is either zero, constant, or depends only on t, and
not on y.
• Nonlinear differential equations are any equations that cannot be
written in the above form. In particular, these include all equations
that include y, y ′ , y ′′ , etc., raised to any power (such as y 2 or (y ′ ) 3 );
nonlinear functions of y or any derivative to any order (such as sin(y)
or e ty ; or any product or function of these.
The order of a differential equation is the degree of the highest order
derivative in it. Thus
y ′′′ − 3ty 2 = sin t (1.5)
is a third order (because of the y ′′′ ) nonlinear (because of the y 2 ) differential
equation. We will return to the concepts of degree and type of ODE later.
Definition 1.1 (Standard Form). A differential equation is said to be
in standard form if we can solve for dy/dx, i.e., there exists some function
f(t, y) such that
dy
= f(t, y) (1.6)
dt
We will often want to rewrite a given equation in standard form so that we
can identify the form of the function f(t, y).
Example 1.1. Rewrite the differential equation t 2 y ′ +3ty = yy ′ in standard
form and identify the function f(t, y).
The goal here is to solve for y ′ :
hence
t 2 y ′ − yy ′ = −3ty
⎫⎪ ⎬
(t 2 − y)y ′ = −3ty
y ′ =
3ty
(1.7)
⎪ ⎭
y − t 2
f(t, y) =
3ty
y − t 2 (1.8)