Analytical Solution: The
Method of Integrating Factor
In this section we are interested in _nding the general solution to the _rst order
linear non-homogeneous equation
y0 + p(t)y = g(t) (3.1)
where p(t) and g(t) are continuous on the open interval a < t
Existence and Uniqueness of
Solutions to First Order Linear
IVP
Solutions to di_erential equations can be given in one of the following forms:
_ by an explicit formula: For example, the function y =
p
t3 + 1 is an explicit
solution to the initial value pr
Existence and Uniqueness of
Solutions to First Order
Nonlinear IVP
When a mathematical model is constructed for physical systems, two reasonable
demands are made. First, solutions should exist if the model is to be useful at
all. Second, to work e_ectivel
Exact Differential Equations
We shall now present another technique for solving _rst order, non-linear, ordinary
di_erential equations. This technique is a generalization of the one we used for
separable equations.
We have seen that the solution procedure
Second Order Linear
Homogeneous Equations with
Constant Coe_cients: Distinct
Characterisitc Roots
In the previous section we discussed the structure of the general solution of a second
order linear homogeneous di_erential equation. As we saw, the general
The General Solution of 2nd
Order Linear Homogeneous
Equations
In this section we discuss the structure of the general solution to the homogeneous
second order linear di_erential equation
y00 + p(t)y0 + q(t)y = 0 (11.1)
where p(t) and q(t) are continuous
Arkansas Tech University
Department of Mathematics
Introductory Notes in Ordinary Di_erential
Equations for Physical Sciences and
Engineering
Marcel B. Finan
March 13, 2014
Preface
Di_erential equations arise from the study of problems in virtually every
Substitution Techniques:
Bernoulli and Riccati Equations
A well-known non-linear equation that reduces to a linear one with an appropriate
substitution is the Bernoulli equation given by
y0 + p(t)y = g(t)yn (7.1)
where n is an integer di_erent from 0 and
Separable Differential
Equations
In this section we discuss an analytical method for solving a type of nonlinear _rst
order di_erential equations, the separable di_erential equations.
A _rst order di_erential equation is separable if it can be written wit
Second Order Linear
Differential Equations:
Existence and Uniqueness
Results
To this point we have considered only _rst order di_erential equations. However,
many of the most interesting di_erential equations involve second derivatives. Indeed,
since acce
Numerical Solutions to
ODEs: Euler's Method and its
Variants
Whenever a mathematical problem is encountered in science or engineering, which
cannot readily or rapidly be solved by a traditional mathematical method, then
a numerical method is usually sough
Practice Problems
Problem 1.1
A car starts from rest and accelerates in a straight line at 1.6 m/sec2 for 10
seconds.
(a) What is its _nal speed?
(b) How far has it travelled in this time?
Problem 1.2
An object is thrown upward at time t = 0: After t seco