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PHYSICS W3003: MATHEMATICAL BASICS
1. Differential equation basics
1.1. Example. Newton’s equation for the motion of a particle,
d2 R(t)
= F(R, t)
dt2
is an example of a differential equation, an equation relating one or more derivatives of
a desired function (in this case the position, R) to something else (in this case the force,
F, acting on the particle). In the usual applications of Newton’s equation the position R
is regarded as a function of time, t; in other words, we want to know how the position
depends on time.
(1)
m
1.2. Classification of Differential Equations.
1.2.1. Ordinary vs partial differential equations. In PHY 3003 we will be concerned with
ordinary differential equations, in which the functions of interest (for example the positions
of particles) depend on a single variable (for example, time). Ordinary differential equations
are to be distinguished from partial differential equations where the functions of interest
depend on several variables. For example in electrodynamics, the strengths of electric and
magnetic fields depend on position and on time and the Maxwell equations that determine
the fields are partial differential equations.
1.2.2. Order. The order of a differential equation is the highest derivative appearing in it.
Thus in its standard form, Newton’s equation, Eq. 1, is a second order equation because
the highest derivative is the second derivative d2 /dt2 . On the other hand, the relation
between velocity v and position R
dR
=v
(2)
dt
is a first order differential equation because the highest derivative that appears in it is the
first derivative.
1.2.3. Linearity. An equation is linear if the variable which is to be solved for appears
only to the first or zeroth power; otherwise it is nonlinear. Newton’s equation, Eq. 1, may
be linear or nonlinear according to the dependence of the force F on R. If the force is
independent of position or depends on the first power of the position Newton’s equation is
linear. In other words, if
(3)
F(R, t) = f (t)
or
(4)
F(R, t) = −kR
or
(5)
F(R, t) = −γ
1
dR
dt
2
PHYSICS W3003: MATHEMATICAL BASICS
or more generally
dR
dt
then Eq. 1 is linear. Note that as Eq. 5 shows, derivatives dont affect the linearity–we just
count the powers of the function.
√
If F has a different dependence, for example F = −cR2 or F = −d R then the equation
is nonlinear.
Linear equations are straightfoward to solve; nonlinear equations are in general not.
(6)
F(R, t) = f (t) − kR − γ
2. Linear Differential Equations: General facts
2.1. Homogeneity. A homogeneous linear equation (more precisely an equation which is
homogeneous of degree 1) contains only first powers of the variables of interest and their
derivatives. An inhomogeneous linear equation also contains terms independent of the
variables of interest. Thus
d2 R
dR
(7)
m 2 +γ
+ kR = 0
dt
dt
is a linear homogeneous equation, whereas
d2 z
= −g
dt2
is not homogeneous because if −g on the right hand side is nonzero the equation contains
a term involving the variable to a power different from 1 (in this case 0).
To each linear inhomogenous differential equation we may associate a homogenous equation obtained by removing all of the terms proportional to the zeroth power of the variable
to be solved for. For example, the homogeneous equation associated with Eq. 8 is
(8)
d2 z
=0
dt2
2.2. Superposition. If R1 (t) and R2 (t) are two functions that satisfy a homogeneous
linear differential equation, then the combination R(t) = c1 R1 (t) + c2 R2 (t) also satisfies
the equation if c1 and c2 are constants (independent of R and t),
If Rinh (t) satisfies an inhomogenous linear differential equation and Rh (t) satisfies the
associated homogeneous equation then the combination Rinh (t) + ch Rh (t) also satisfies the
inhomogeneous equation for any constant value of ch .
(9)
2.3. Boundary conditions. Mathematicians have shown that an nth order homogeneous
linear differential equation admits n linearly independent solutions (is solved by n different
functions not related by multiplication by a contstant), so the general solution is f (t) =
P
i=1...n ci fi (t). Determining the solution that is relevant to a particular situation requires
n pieces of information: the constants c1 ...cn . This information is typically presented as
the values of the function and/or its derivatives on the boundaries of the region of interest
(e.g. at some initial time) and the values are often referred to as “boundary conditions”.
Newton’s equation is a second order differential equation, so the solution is uniquely
specified by two boundary conditions. These are often chosen as initial conditions, namely
the values of the position and velocity at some initial time, but might be chosen as terminal
conditions (value of position and velocity at some later time) or as a mixture (for example
the value of the position at an initial time and at a later time).
PHYSICS W3003: MATHEMATICAL BASICS
3
For example, the motion of a particle in the absence of any forces is decribed by
d2 z
=0
dt2
This is a second order, homogeneous linear equation. The two linearly independent solutions can be chosen to be z(t) = t and z(t) = 1. The two parameter family of functions
(10)
(11)
z(t) = v0 t + z0
solves Equation 10 for any values of v0 and z0 . Selecting which member of the family of
functions is relevant to a specific problem requires two additional pieces of information.
One possible choice is to specify the value and first derivative (position and velocity) at
some time, for example t = 0. Another possible choice would be to specify the position at
two different times.
3. Solutions of some standard homogeneous linear differential equations
Here are a few basic linear homogeneous equations whose solution you should know:
• nth derivative only
dn R
=0
dn t
(12)
Solution:
(13)
R(t) =
n−1
X
am tm
m=0
The n coefficients am (m = 0...(n − 1) are fixed by boundary conditions. For
example, if n = 2 Eq. 12 is Newton’s equation for a free particle. The general
solution is
(14)
R(t) = R0 + V0 t
and the two constants R0 and V0 may be fixed by choosing the as boundary conditions the position and velocity of the particle at time t = 0. We could alternatively
fix the position at time t = 0 and at time t = tlater . We would then fix the
parameters R0 and V0 by solving the equations
R(t = 0) = R0
R(t = tlater ) = R0 + V0 tlater
implying R0 = R(t = 0) and V0 = (R(t = tlater ) − R0 )/tlater .
• Exponential
dR
− kR = 0
dt
(15)
Solution:
(16)
R(t) = aekt
Eq. 15 is a 1st order equation so the solution involves 1 coefficient (a) which is fixed
by a boundary condition, for example the value of R at some specified time.
4
PHYSICS W3003: MATHEMATICAL BASICS
• Oscillation
d2 R
+ k2 R = 0
dt2
(17)
Solution:
(18)
R(t) = asin (kt) + bcos (kt) = a+ eikt + a− e−ikt
Eq. 17 is a 2nd order equation so we have 2 independent functions with coefficients
(a and b to be fixed by a boundary condition.
Here we have implicitly assumed k 2 > 0 but if k 2 < 0 then instead of oscillating
solutions we will have growing or decaying solutions.
4. Inhomogeneous equations
Write a general inhomogeneous linear differential equation as
(19)
DR(t) = f (t)
Here DR stands for any linear combination of R and its derivatives (for example the left
hand sides of any of the equations in the section above), and f is some function independent
of R.
The homogeneous equation corresponding to to Eq. 19 is
(20)
DR(t) = 0
To construct the general solution of Eq. 19 one must first find a particular solution: any
function RP (t) that satisfies Eq. 19. To this one should add the most general solution
RH (t) of the corresponding homogeneous equation Eq. 20. The general solution is then
(21)
R(t) = RP (t) + RH (t)
RH (t) will involve constants which may be chosen to ensure that the solution fits the
boundary condition.
For example, Newton’s equation for a particle moving in tine-independent force f is
d2 R
=f
d2 t
You can verify that the particular solution RP (t) = 12 f t2 satisfies Eq. 22. Combining this
with the general solution of the homogeneous equation (given above in Eq. 14) gives the
general solution of Eq. 22 as
1
(23)
R(t) = f t2 + R0 + V0 t
2
with R0 , V0 fixed by boundary conditions.
(22)