estudian

Elementary differential equations
Notice: this material must not be used as a substitute for attending
the lectures

1

1

Differential Equations

An ordinary differential equation, often abbreviated to ODE, is an equation
containing an unknown function and some of its derivatives. It has to be solved to
find the unknown function.
Examples of ODEs are
dy
= cos x,
dx

dy
y
=1− ,dx
x

y − 2y + 2y = 0

where y = dy/dx and y = d2 y/dx2 . In each of these equations y is the unknown
function and the idea is to find y in terms of x.
We shall shortly explain some of the methods available for solving simple ODEs,
but first let us introduce some physical problems which give rise to ODEs.

1.1

Physical problems giving rise to ODEs

(i) Imagine water draining out froma large cylindrical tank through a small hole or
tap at the bottom. Then the depth of water remaining in the tank is constantly
changing, call it h(t) where t is time. From the theory of fluid mechanics
(Bernoulli’s equation in particular) it can be shown that h(t) must satisfy the
differential equation
√
dh
= −k h
dt
where k is some constant depending on the tap radius, the tank radius andthe
acceleration due to gravity.
(ii) The differential equation for a simple pendulum of length l is
d2 θ g
+ sin θ = 0
dt2
l
which is to be solved for the angle θ(t).
(iii) Let y (x) be the equation for the curve described by a heavy chain or rope hanging under gravity. It can be shown that y (x) is the solution of the differential
equation
d2 y
dy
=k 1+
dx2
dx

2

You mayalready be aware that the curve in question is called a catenary and
is described by the hyperbolic function cosh.
(iv) The differential equation
m

d2 s
ds
− (µα − β )
dt2
dt

2

2

= T − µmg

has been used to model an aircraft taking off. Here, s = s(t) is the distance
travelled along the runway at time t, m is the aircraft’s mass, g is acceleration
due to gravity, T is the thrustfrom the engines, µ is the coefficient of friction,
β is called the aerodynamic drag parameter and α is the lift parameter. The
idea is to solve the differential equation to find s as a function of t.

1.2

Basic terminology

The order of a differential equation is the order of the highest order derivative appearing. For example,
dy
+ 2y = 1 is first order
dx
dy
dx

2

+ 2y = 1

is firstorder

d2 y
+ ω 2 y = sin 2t is second order
dt2
dy
y
= t2 is first order
dt
d3 y
− y = 0 is third order
dx3
A solution of a differential equation is a function that satisfies it for all values of the
independent variable.
dy
For example y = Ce2x is a solution of dx = 2y . We can verify this by substituting
in. When y = Ce2x ,
dy
= 2Ce2x = 2y
dx
and therefore the differentialequation is satisfied.
dy
Another example: y = x2 − 5x satisfies x dx − y = x2 . We shall again check this
dy
by verification. When y = x2 − 5x, dx = 2x − 5 and so
x

dy
− y = x(2x − 5) − (x2 − 5x) = x2
dx

i.e. the differential equation is satisfied.
Of course, this is not how we normally “solve” differential equations. We do not
simply pluck a function out of thin air and then show that itfits. We want systematic
ways of actually finding the solution. There are many different methods available,
and the appropriate method depends on what kind of differential equation you are
dealing with. The first kind of differential equation we shall treat is called a separable
differential equation.

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Separable equations

If a differential equation can be put into the form
g (y ) dy = f(x) dx
then it is said to be separable. This is because the variables are separated, all y ’s on
one side and all x’s on the other. To solve such an equation we integrate both sides:
g (y ) dy =

f (x) dx + c

and then get y in terms of x if possible.
Not all first order equations can be put into this form. You need to be able to
recognise those that can. It is a question of building up...