1 Introduction
You have already studied ordinary differential equations (ODEs) and have learnt how to obtain the solution of certain types. Since a knowledge of the solution of certain ODEs (i.e. those with constant coefficients) will be required in solving partial differential equations (PDEs), we will begin this unit reminding you of some important results.
Key Point 1
The first order ODE
In Key Point 1 the quantity in the general solution is a constant. To obtain the value of we have to know the value of at some value of , perhaps . In other words, we need to know an initial condition .
Task!
Find as a function of if
and the initial condition is
From Key Point 1 with we have the general solution
Putting and into this we obtain i.e. so the solution to the given initial value problem is
We shall also need to be familiar with solutions to second order, homogeneous, constant coefficient ODEs, summarised in Key Point 2.
Key Point 2
A second order ODE of the form
(1)
where are constants, has an auxiliary equation
(2)
obtained by inserting the trial solution in (1).
The general solution of (1) then depends on the solutions (or roots) of the quadratic equation (2).
-
If (2) has real, distinct roots
and
then
-
If (2) has a repeated root
then
-
If (2) has complex roots (which will be a conjugate pair)
then
Note that in each of these cases 1. to 3. the general solution is a linear combination of two particular solutions:
For 1. they are and .
For 2. they are and .
For 3. they are and .
Task!
Use Key Point 2 to find the general solution of
First write down the auxiliary equation:
Now find the roots of the auxiliary equation:
Finally give the general solution to the ODE:
(Since the roots of the auxiliary equation are real and distinct.)
Task!
Find the general solution of
First write down the auxiliary equation:
Now Find the roots of this auxiliary equation:
Finally give the general solution to the ODE:
(Since the roots of the auxiliary equation are complex conjugates with real part and imaginary part .)
The two Tasks above can be generalised as in Key Point 3.
Key Point 3
-
The general solution to:
is
-
The general solution to:
is
Those of you who are familiar with elementary dynamics will recognise the second differential equation in Key Point 3 as modelling simple harmonic motion .