ES 111 Mathematical Methods in the Earth Sciences
Lecture Outline 16 - Weds 19th Nov ’08
Higher Order Differential Equations
Here we are going to move on the higher-order differential equations (meaning derivatives higher
than first order may appear). This is a big field, so we will focus mainly on solving specific, relatively
simple higher order ODE’s which often crop up in Earth Sciences.
When are they useful? Especially in the equations of elasticity (e.g. seismic waves) and
flexure.
The general form of a n-th order linear ODE is
an(x)y(n)+an−1(x)y(n−1) +· · · +a1(x)y0+a0(x)y=f(x)
where the an(x) and f(x) are functions of x. Written in this form f(x) is known as the input or
driving term (e.g. in elasticity, it represents the load being applied). A linear ODE with f(x)=0
is called homogeneous. Thus, a homogeneous linear ODE (HLDE for short) has no driving term.
Here we will focus on HLDE’s.
Solutions to HLDE’s
In general, there is more than one solution to a HLDE. Suppose that y1(x) and y2(x) are both
solutions of a HLDE, then any linear combination of them
c1y1(x) + c2y2(x)
is also a solution. This is true only of HLDE’s - if the driving term is not zero, then this ability to
superpose solutions will not in general be true.
The general solution to an n-th order HLDE is constructed by a linear combination of nlinearly
independent solutions to the HLDE. This is similar to the way we can construct any vector from a
linear combination of the perpendicular vectors ˆ
i,ˆ
jand ˆ
k.
A bunch of functions are linearly independent if and only if the only solution to
c1y1(x) + c2y2(x) + · · · +cnyn(x) = 0
is c1=c2=· · · =cn= 0 for some xin the range of interest. In other words, none of the functions
can be constructed from a linear combination of the other functions.
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