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Unformatted text preview: Section 1.1: Classiﬁcation of Diﬀerential
Equations; Their Origins and Applications Deﬁnition 1 An equation with derivatives with dependent and independent variables is a diﬀerential equation.
Below are examples:
dy
d2 y
+ xy ( )2 = 0
2
dx
dx
Derivative identities are not considered diﬀerential equations.
Deﬁnition 2 A diﬀerential equation with ordinary derivatives of
one or more dependent variables with respect o on independent variable is an ordinary diﬀerential equation ⇒ ODE
d4 x
d2 x
+ 5 2 + 3x = sint
dt4
dt
Deﬁnition 3 A diﬀerential equation with partial derivatives of one
or more dependent variables with respect to one or more independent
variables is called a partial diﬀerential equation ⇒ PDE
∂v ∂v
+
=v
∂s
∂t
∂ 2v ∂ 2u ∂ 2u
+
+
=0
∂x2 ∂y 2 ∂z 2
Deﬁnition 4 The order of the highest ordered derivative in a differential equation is the order of the diﬀerential equation. 1 d3 y d2 y
dy
+ 2 + 5 + 6y 2 = 0
3
dx
dx
dx
The Diﬀ Eq above has an order of 3.
Deﬁnition 5 A linear ODE of order n, in the dependent variable
and the independent independent variable is an equation that can be
expressed in the form:
dn y
dn−1 y
dy
+ a1 (x) n−1 + . . . + an−1 (x) + an (x)y = b(x)
n
dx
dx
dx
Below are examples of linear ODE
a0 (x) d2 y
dy
+ 5 + 3y = 0
2
dx
dx
d4 y
d3 y
dy
+ x2 3 + x3
= xex
4
dx
dx
dx
Deﬁnition 6 A nonlinear ODE is ODE that is not linear
d2 y
dy
+ 5 + 6y 2 = 0
2
dx
dx
d2 y
dy
+ 5( )3 + 6y = 0
2
dx
dx
dy
d2 y
+ 5y
+ 6y = 0
2
dx
dx
Linear ODE are further classiﬁed by the nature of coeﬃcients of
dependent variables and derivatives
The equation below is linear with constant coeﬃcients:
d2 y
dy
+ 5 + 6y = 0
2
dx
dx
The equation below is linear with variable coeﬃcients:
d4 y
d3 y
dy
+ x2 3 + x3
= xex
4
dx
dx
dx
There are many Diﬀerential Equation problems in science and engineering
2 ...
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 Spring '11
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