Math 201 — Differential Equations (2.1)
Chi-Kun Lin
Department of Applied Mathematics
National Chiao Tung University
Hsin-Chu 30010, TAIWAN
3rd September 2009
Chi-Kun Lin
Math 201 — Differential Equations (2.1)
Newton’s 2nd law
Newton’s 2nd law F = ma is an ordinary differential equation!
Question: How to solve this differential equation?
Answer: Isaac Newton created calculus to solve D. Es.
x(t2 ) − x(t1 )
t2 − t1
avergare speed
dx
4x
x(t2 ) − x(t1 )
≡ lim
= lim
t2 →t1
4t→0 4t
dt
t2 − t1
instaneous speed
I know not what I appear to the world, but to myself I
seem to have been only like a boy playing on the
sea-shore, and diverting myself in now and then finding a
smoother pebble or a prettier shell, whilest the great
ocean of truth lay all undiscovered before me.
— Isaac Newton (1643 - 1727) —
Chi-Kun Lin
Math 201 — Differential Equations (2.1)
Newton’s 2nd law (continue)
Fundamental units (Recall cgs-system or MKS-system)
unit of length = cm/L,
unit of mass = g/M,
unit of time = s/T,
{L, M, T}: fundament units
v=
a=
dx
: speed
dt
d 2x
dv
=
: acceleration
2
dt
dt
[v ] =
h dx i
[a] =
dt
= LT−1
h d 2x i
dt 2
=
h dv i
dt
= LT−2
It is customary, following a suggestion of J. C. Maxwell
(1831-1879), to denote the dimension of a quantity f by [f ].
Chi-Kun Lin
Math 201 — Differential Equations (2.1)
Newton’s 2nd law (continue)
Definition
an
dy
d ny
d n−1 y
+ a0 y = 0,
+
a
+ · · · + a1
n−1
n
n−1
dx
dx
dx
an 6= 0
is a n-th order ordinary differential equation.
F = ma
⇐⇒
F = ma = m
dv
dt
is a 1st order differential equation of v (speed).
F = ma
⇐⇒
F = ma = m
d 2x
dt 2
is a 2nd order differential equation of x (position).
Chi-Kun Lin
Math 201 — Differential Equations (2.1)
Motion of a falling body
F = FG − FR = mg − bv
[mg ] = [bv ]
=⇒
− : opposite direction
MLT−2 = [b]LT−1
=⇒
[b] = MT−1
dv
= mg − bv
dt
Question: How to solve this equation?
Method: separation of variables (v to v and t to t)
F = ma
=⇒
m
Give to Caesar what is Caesar’s, and to God what is
God’s.
— Bible —
Chi-Kun Lin
Math 201 — Differential Equations (2.1)
Motion of a falling body: separation of variables
dt
dv
=
mg − bv
m
b>0
To solve the differential equation is reduced to the problem of
integration.
Need: technique of integration!
Z
Z
dv
dt
=
(indefinite integral)
mg − bv
m
t
t
1
+ C,
[C ] =
= TM−1
− ln |mg − bv | =
b
m
m
A bt
mg
− e − m , A = e −bC , [A] = [mg ] = MLT−2
b
b
general solution! (A is still not determined.) The constant A or C
is determined by initial condition.
v (t) =
Chi-Kun Lin
Math 201 — Differential Equations (2.1)
Motion of a falling body: initial value problem
m
dv
= mg − bv ,
dt
General solution: v (t) =
v (0) =
mg
b
v (0) = v0
bt
− Ab e − m
A
mg
− = v0
b
b
=⇒
A = mg − bv0
Solution of the initial value problem (I.V.P.)
v (t) =
mg
mg − bt
+ (v0 −
)e m
b
b
v∞ ≡ lim v (t) = lim
t→∞
t→∞
mg
mg − bt
+ (v0 −
)e m
b
b
Chi-Kun Lin
=
mg
b
Math 201 — Differential Equations (2.1)
Motion of a falling body: limiting (terminal) velocity
v∞
mg
=
,
b
mg
b
=
MLT−2
L
=
−1
MT
T
v∞ is independent of the initial velocity v0 ! Assuming similar
shape,
m1 > m2
=⇒
The heavier, the faster!
v1,∞ > v2,∞
The limiting speed v∞ is predictable from the equation.
Need
dv
=0
dt
dv
= mg − bv =⇒ 0 = mg − bv∞ =⇒
dt
is obtainable without solving the equation.
m
v∞
lim
t→∞
Chi-Kun Lin
v∞ =
Math 201 — Differential Equations (2.1)
mg
b