MA2001N Differential Equations
Lecture Notes for Week 2
[3. 2nd order, linear odes, reducible
to 1st order form]
[4. 2nd order odes with constant
coefficients]
3.
2nd order, linear odes, reducible to 1st order form
3.1
The general solution
Consider the 2n
MA2001N Differential Equations
Example Sheet 8
Examples of Solutions by Laplace Transform methods
In the questions that follow, apply the solution procedure as outlined in section 8 of the
notes for Laplace Transforms. The Worked Examples (a), (b) and (c)
MA2001N Differential Equations
Lecture Notes for Week 1
[1. Revision of terminology]
[2. Revision of the solution
of 1st order odes]
1.
Revision of terminology
1.1
What is an Ordinary Differential Equation (or ODE)?
An ordinary differential equation (ode)
MA2001N Differential Equations
Application Problems 2: Mass, spring and damper problems
(of homogeneous form)
Many problems in mechanics, such as, a weighing machine or a car suspension system, can
be modelled by considering the sum of forces due to a mas
MA2001N Differential Equations
Example Sheet 7
Examples of Solutions in Series about an Ordinary Point
In the questions that follow, apply the general solution procedure, as outlined in section 7
of the notes for weeks 7 and 8. The Worked Examples (a) and
MA2001N Differential Equations
Example Sheet 5
Reduction of Order III: Non-homogeneous Form Examples
In the questions that follow, use the Reduction of Order formula for y P , that is,
W
y P y1 2
y1
y1 g ( x)
W dx dx
(A)
where
W exp(
p( x) dx)
,
to fin
MA2001N Differential Equations
Application Problems 1: Population Dynamics
It is possible to set up a mathematical model of the behaviour of a population,
as its size, P , changes with time, t . Problems in the growth, or the decay, of
the population, P (
MA2001N Differential Equations
Application Problems 3: Mass, spring and damper problems
(of non-homogeneous form)
As we have seen previously, in Application Problems 2, many problems in mechanics, such as,
a weighing machine or a car suspension system, ca
MA2001N Differential Equations
Example Sheet 4
Reduction of Order II: Homogeneous Form Examples
In the questions that follow, use the Reduction of Order formula for y 2 , that is,
W
y 2 y1 2 dx ,
y1
(A)
where
W exp(
p( x) dx)
,
to solve the given ode in c
MA2001N Differential Equations
Answers to Example Sheet 1:
Revision Sheet
Note that, in all answers given below, k is a constant.
(a) Revision of Integration Techniques
(i)
Integration using Partial Fractions
The partial fractions and the integral for eac