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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16 3 If we did not keep track of the initial condition y (0) = y0 , we would call the solution of the diﬀerential
equation alone the general solution. For example, y (t) = (b/a) + C1 eat would be the general solution for the
diﬀerential equation above because it involves an arbitrary constant C1 . If we do keep track of the initial
condition, we would call the solution the particular solution. For example, y (t) = (b/a) + (y0 − b/a) eat would
be the particular solution corresponding to y (0) = y0 .
Classification of Differential Equations
Types of Diﬀerential Equations. There are two:
• ODEs or ordinary diﬀerential equations are diﬀerential equations which do not involve partial derivatives.
• PDEs or partial diﬀerential equations are diﬀerential equations which do involve partial derivatives.
Each of the examples of diﬀerential equation we have seen are types of ODEs, so here is an example of a
PDE: Say that we have a rod of length L sitting in or above a heat source. (For example, the rod might have
one end in a furnace.) We wish to compute the temperature u = u(x, t) as a position x (where 0 ≤ x ≤ L)
and a time t. Note that both time t and position x are independent variables, and u = u(x, t) is a dependent
variable. The diﬀerential equation which models this physical situation is called the Heat Equation :
∂2u
∂u
=
∂ x2
∂t
for some nonzero constant α which depends on the characteristics of the rod.
In this course, we will only be interested in ordinary diﬀerential equations.
α2 Order of an ODE. Consider a function y = y (t). Let y (k) denote the k th derivative. For example,
dy
d2 y
,
y (2) = 2 ,
dt
dt
An ordinary diﬀerential equation is an expression in the form
F t, y, y (1) , y (2) , . . . , y (n) = 0
y (0) = y, y (1) = ··· for some function F . The largest order n of derivative y (n) which appears in this expression is called the
order of the diﬀerential equation. We give a few of examples.
Let x = x(t) denote the position of a mass m at time t under the inﬂuence of gravity and air resistance.
If this mass is near sea level (at the surface of the Earth), Newton’s Law of Gravity states dx
d2 x
+ m 2 = 0.
dt
dt
This is an ordinary diﬀerential equation of second order because it involves the second derivative of the
position x = x(t). If instead we use the velocity v = x (t), we ﬁnd the diﬀerential equation
dv
mg + γ v + m
= 0.
dt
This is an ordinary diﬀerential equation of ﬁrst order because it involves just the ﬁrst derivative of the
velocity v = v (t).
Let P = P (t) denote the size of a population at time t. If we assume that the rate of change of the size
if proportional both to the size of the population as well as the diﬀerence from the maximal sustainable size
K of the population, then we ﬁnd the diﬀerential equation
P
dP
−r P 1 −
+
= 0.
K
dt
This is an ordinary diﬀerential equaiton of ﬁrst order because it involves just the ﬁrst derivative of the
population size P = P (t).
mg + γ ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins

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