ODE Lecture Notes
Section 1.3
Page 1 of 4
Section 1.3: Classification of Differential Equations
Big Idea: There are a ton of different kinds of differential equations, and they have been grouped
and classified according to their methods of solution.
Big Skill: You should be able to identify some of the basic classifications of differential
equations, and verify a given solution of a differential equation.
Ordinary vs. Partial Differential Equations
Ordinary Differential Equations
 Only ordinary derivatives appear in the
equation
Example:
d 2Q  t 
dQ  t  1
L
R
 Q t   E t 
2
dt
dt
C
(RLC series circuit with a drive voltage)
Partial Differential Equations
 Only partial derivatives appear in the
equation
Example:
2
u  x, t 
2  u  x, t 


2
x
t
(heat conduction equation)
Systems of Differential Equations
 …consist of multiple differential equations involving multiple variables.
 Used when one quantity depends on other quantities…
dx
 ax   xy
dt
 Example: Lotka-Volterra (predator-prey) equations:
dy
 cy   xy
dt
Order of a Differential Equation
 …is the order of the highest derivative that appears (analogy: the degree of a polynomial
is the largest exponent)
n
 General form for an nth order differential equation: F t , u  t  , u  t  , , u    t   0 for



some function F.
We’ll deal with differential equations that can be solved explicitly for the highest
n
n 1
derivative: y    t   f t , y  t  , y  t  , , y    t  .


ODE Lecture Notes
Section 1.3
Page 2 of 4
Practice:
1. Put the third-order differential equation F  t , y  t  , y t  , y t  , y t    0 into standard
form given F  s, x0 , x1 , x2 , x3   x3  2es x2  x0 x1  s 4 .
Linear vs. Nonlinear Differential Equations
 F t , y  t  , y  t  , , y  n  t   0 is linear if F is a linear function of y  t  , y  t  ,





, y
n
t  ;
note: F does not have to be linear in t…
The general form of a linear differential equation of order n is:
n
n 1
an  t  y    t   an1  t  y    t    a1  t  y  t   a0  t  y t   g t 
Nonlinear differential equations are not of the above form…
Many times, nonlinear equations can approximated as being linear over small regions of
the solution. This process is known as linearization.
Practice:
2. Derive the nonlinear differential equation for a simple pendulum using the torque
equation   I , then linearize the equation.
ODE Lecture Notes
Section 1.3
Page 3 of 4
Solutions of Differential Equations
 …are functions  such that all necessary derivatives exist over an interval of interest of
the independent variable and that satisfy   n  t   f t ,   t  ,    t  , ,   n1  t  .



To verify a given solution, substitute it and its derivatives into the given differential
equation to show that the equation is satisfied.
Practice:
 g 
 g 
t   B cos 
t  is a general solution of the linearized
3. Verify that   t   A sin 
 L 
 L 
simple pendulum problem.
4. Verify that the second-order differential equation t 2 y  5ty  4 y  0 has a general
solution of the form y  t   At 2  Bt 2 ln t .
ODE Lecture Notes
Section 1.3
Page 4 of 4
5. Verify that the second-order partial differential equation  2u xx  ut has a general solution
of the form u  t   Ae t sin x  Be 
2
 t
2 2
sin  x for  being any real constant.
6. Assuming that the third-order differential equation y  3 y  2 y  0 has a solution of the
form y  ert , find values of r that satisfy the equation.
7. Assuming that the differential equation t 2 y  4ty  4 y  0 has a solution of the form
y  t r , find values of r that satisfy the equation.