Unit I Lesson 1
Definitions and classification
Definition: A differential equation is any equation that contains a derivative
Ex1 The equation
dy
dx
2
+ y = cos( x) +
d y
contains a first and a second derivative.
2
dx
3
Ex 2 x⋅ y'' − x ⋅ y''' = sin( y) contains a second and third derivative.
Classification of differential equations
Differentiual equations are classified by:
A. order: the highest derivative appearing in it.
B. linear and non-linear: If the function and its derivatives only appear to the power 1 and
not as a part of some other function, then it is linear.
C. ordinary or partial: a partial differential equation (PDE) has at least one partial derivative
in it. An ordinary differential equation (ODE) has only ordinary derivatives in it.
Ex 3 the4 orders of Ex 1 and Ex 2 above are 2 and 3 respectively.
Ex 4 Here is a PDE:
∂x
∂z
2
+ 5x ⋅ y =
2
Ex 5 This one is non-linear
∂y
∂z
 dy  + 7y = x3y because of the term
 dx 
 
 dy 
 dx 
 
2