Math 150 (Lecture 4)
Separable Equations
Denition 1. A equation is said to be separable if it is written in the following form
M (x) + N (y )
dy
= 0.
dx
(1)
Example 1. Determine whether the following equation is separable:
(a)
dy
2x
=
,
dx
y + x2 y
(b)
x2
Math 150 (Lecture 3)
Linear Equations; Method of Integrating Factors
Denition 1. The general rst order linear equation is of the following form:
dy
+ p(t)y = g (t)
dt
where p and g are given functions of the independent variable t.
Example 1. Solve the di
Math 150 (Lecture 5)
Modelling with First Order Equations
Dierential equations are of interest to non-mathematicians primarily because of the possibly of using them to investigate a wide variety of problems in the physical, biological, and
social sciences
Math 150 (Lecture 6)
Autonomous Equations and Population Dynamics
Denition 1. A rst order equation is called autonomous if it is of the following form
dy
= f (y ).
dt
(1)
where f (y ) does not depend on t.
Exponential Growth
Let y = (x) be the population
Math 150 (Lecture 7)
Exact Equations
Example 1. Solve the dierential equation
2x + y 2 + 2xyy = 0.
The equation is neither linear nor separable, so the methods developed before cannot
apply here. The way of solving this equation relies on a key observatio
Math 150 (Lecture 8)
Homogeneous Equations with Constant Coecients
The general form of the second order linear dierential equations is
P (t)
d2 y
dy
+ Q(t) + R(t)y = G(t)
2
dt
dt
(1)
where P (t), Q(t), R(t) and G(t) are given functions. Usually, we rewrit
Math 150 (Lecture 9)
Repeated Roots
Example 1. Solve the dierential equation
y + 4y + 4y = 0.
(1)
The characteristic equation is
r2 + 4r + 4 = (r + 2)2 = 0.
So r1 = r2 = 2. Hence, one solution is y1 (t) = e2t . To nd the second one, we try to
nd a functio
Math 150 (Lecture 10)
Fundamental Solution
The following Theorem is the fundamental result for second order linear dierential equations.
Theorem 1. Consider the initial value problem
y + p(t)y + q (t)y = g (t),
y (t0 ) = y0 ,
y (t0 ) = y0 ,
where p(t), q
Math 150 (Lecture 11)
Nonhomogeneous equations
In this lecture, we turn to the nonhomogeneous equation
y + p(t)y + q (t)y = g (t),
(1)
where p, q and g are given continuous function on the open interval I . The equation
y + p(t)y + q (t)y = 0,
(2)
in whic
Math 150 (Lecture 12)
Variation of Parameters
In this lecture we describe another method of nding particular solution of a nonhomogeneous equation. This method, known as variation of parameters, is a general method,
i.e., it can be applied to any equation
Math 150 (Lecture 13)
Mechanical and Electrical Vibration
Spring-mass system: Consider a mass m hanging on the end of a vertical spring of
original length l. The mass causes an elongation L of the spring in the downward (positive)
direction. There are two
Math 150 (Lecture 14)
Forced Vibration
We now study the case where a periodic external force is applied to a spring-mass system.
Forced Vibration with Damping:
Suppose that the external force is given by F0 cos t, where F0 and are positive constants
repre
Math 150 (Lecture 15)
The Laplace transform
Denition 1. The Laplace transform of f (t), which we will denote by Lf or by F (s), is
dened by
est f (t)dt.
Lcfw_f (t) = F (s) =
(1)
0
Observe that we take integration over an unbounded interval , (0, ), is cal
Math 150 (Lecture 16)
Step functions
Denition 1. Step function is denoted by uc (t), and is dened by
0, 0 t < c
1,
c t,
uc (t) =
c 0.
(1)
Example 1. Represent the function
0, 0 < t <
1, t < 2,
h(t) =
0,
2 t,
by step functions.
The Laplace transform of u
Math 150 (Lecture 17)
Discontinuous Forcing Functions
Now we turn our attention to some examples in which the nonhomogeneous term, or forcing
function, is discontinuous.
Example 1. Find the solution of the dierential equation
2y + y + 2y = g (t),
where
g