257
I ntern. J. Math. & Math. Sci.
Vol.
(1978)257-267
NONLINEAR DIFFERENTIAL EQUATIONS AND ALGEBRAIC SYSTEMS
LLOYD K. WILLIAMS
Department of Mathematics
Texas Southern University
Houston, Texas 77004
U.S.A.
(Received April 17, 1978)
ABSTRACT.
In this paper we obtain the general solution of scalar, first-order
differential equations.
The method is variation of parameters with asymptotic
series and the theory of partial differential equations.
The result gives us a form like a differential quotient requiring only
that a limit be taken.
Like the familiar expression for the solution of linear,
first order, ordinary equations, it is the same in all cases.
KEY WORDS AND PHRASES. Riccati Equation, Abel Equatns, Cauchy-Kowalewski
Theorem, Cauchy-Kowalski System, Unival Cauchy-Kowalski System.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
i.
34AOS, 34A10, 34A15.
INTRODUCTION.
We present a unified treatment for the general scalar, first-order, ordinary
differential equation
y’
G(x,y)
G e C
I
L. K. WILLIAMS
258
Particular examples are linear equations, Riccati equations and Abel
equations.
2.
PRELIMINARIES.
We begin with the differential system
IVI’
V
VIV 2
f(VI,V2)
2’ h(VI,V2)
V
V
1
(2.1)
2
v#0
with general solution V
l(x,c I,c2),
V
I
V
V 2(x,c l,c 2).
2
Here
Cl,C 2
are
arbitrary constants.
Now let x
x(t).
Then we get
ddt
i UI
V
U
2
U21
1
f(Vl,V2)
(2.2)
U
2
h(Vl,V2)
We are now ready to present the algebraic system referred to in the title.
3.
THE CAUCHY-KOWALEWSKI SYSTEM.
Let wI
Wl(t,e)
w
2
w2(t,e)
be two functions of t and e (at present
unknown).
VI,V2 have been given by (2.1). Finally
K(Wl,W2,t,e) and L(Wl,W2,t,e) will be defined by
The functions
functions
ential equations later.
two more unknown
partial differ-
They will contain another variable, %.
possible to substitute an arbitrary
G(Wl,t)
It will be
for % to solve specific equations.
NONLINEAR DIFFERENTIAL EQUATIONS
DEFINITION.
259
The system of algebraic equations
(a)
w
(b)
w
(c)
x
K(Wl,W2,t,e)V I
I
L(Wl,W2,t,e
w
I
+
tw
0
V
2
0
2.
Wl#0
is called the Cauchy-Kowalewski system, for a specific
G(Wl,t).
Using % we
will get a universal system.
Under suitable conditions on the functions K and L, we can solve it for
w
I
Wl(t,e)
and w
w2(t,e).
2
We proceed by defining these functions as
solutions of appropriate partial differential equations.
these functions
L(Wl,W2,t,e,%
like universal constants.
4.
THE
FIRS
and
-
K(Wl,W2,t,e,%
We will derive
and regard them as fixed
FUNCTION K IN THE CAUCHY-KOWALEWSKI SYSTEM.
We differentiate 3(a-b) with respect to t to get expressions for
Denoting the expression for
To simplify notation, let K
i
Some of the
derivatives.
.
Now let
h
+
Ai,
i
1
R
(4.)
e in (4.1) and get
AIL2 + A2L 3 + A3
R
_A2LI + A4L2
+
(4.1a)
A5
i,...,5 are given explicitly later.
These are not partial
By contrast
LI
L-
i’2"
by R we get
w and note that
z in the new notation.
L
w--
etc.
from (2.1), 3(a-b) we have f
L. K. WILLIAMS
260
We arbitrarily set
The following equation is of fundamental importance.
where K
a
tf 2
2WlW2eI witha I + Wl 3 +
A2
(L,Wl,W2,t,e)
and e
-,etc.,
I
for real e > 0.
By the Cauchy-Kowalewski theorem [See e.g. (2.1)] let
be an analytic solution of (4.2).
A
Let
o n=0l Cne
conditions on c
O
DEFINITION.
o
)re
where
are analytic.
we give the following definitions.
w
1
of the A
[(---+ z)(wlo4 wlW2ao2 + w2o)
L
ao(L’wl,w 2,t,e)
1,2,3,4,5.
i
Cn(L,w1,w2,t)
Cn
eo
Further, we will write
Ai(a o),
i
(4.2)
e
Before imposing
Si(L,wi,w2,t
will now be given explicitly.
i
w
1
(--+
z)(Wleo4 WlW2So2 + W2ao)
O
o
Wlmo2 + a2f
A
4
DEFINITION.
L A
-0 I
L
DEFINITION.
(z)
= /o
0
Substituting
o
O
G(w
1
l,t)A--4)
S
2(L,wl,w2,t).
can be stated now as follows:
(2)
S
7.
c
n=0
Wlhm
o
A.
(--
The conditions on c
m
I(L,wl,w2,t) $
n
in
0,
(3)
a#o.
(4.2) we get
w
1
Z)Col + WlC03 + tWlZC o
2WlW2Col- Wlt(--+
O
--0
(4.3)
NONLINEAR DIFFERENTIAL EQUATIONS
261
of which some solutions are given
+ I
H[8(Co,Z,Wl),W2
P(Co,Wl,8(Co,Z,Wl))
(4.3a)
constant
where
(i)
H
is arbitrary
(2)
8
satisfies the partial differential equation
w
(3)
P
+
(--O + z)82
(wi8 3
+
Co81
0
0)
is defined as follows:
Q(Co,wl,a ).
for z
I
CoZSl
first solve
8(Co,Wl,Z)_
a
Then set
d c
CoQ (Co,wl,a)
THEOREM i.
conditions
PROOF.
The function H can be chosen analytic in (4.3a) so that
(2.1), (2.2), (3.1) hold for cO
Let y
i
w
2
+P
and then (4.3a) becomes H(B,y)
constant.
The
partial derivatives of c o are computed from (4.3a) and from them we see that
H
+
0 implies that
implies
merely
w18 3 +
cO
-i--on
( + w2)HY
Hy
Col
0.
Now S
0.
1
+ 0,
+
+
So H
0 implies A
Y
0 implies that
tw2(wlB 3 + CoBl)H8
0, we can choose H so that S
Summarizing the results of this section, K
solution of (4.2) where H is analytic
(3.1) however, we
5.
c
O
0
e
S
+
H
0.
Since
This completes the proof.
0.
l
(2.2) hold if
Thus (2.1)
0.
L-AI
+0
Further, A
so condition (2.i) holds.
1
e
0
O
can be defined as the
and A
0
To solve
must define L.
SOLUTION OF THE CAUCHY-KOWALEWSKI SYSTEM.
To solve the system (3.1), we must now define the function
L(Wl,W2,t,e).
0
L.K. WILLIAMS
262
o
G,
Setting
and A
2
,
(4.2) in (4.1a) suggests defining L
by
GL
L
I
L
I
8w
1
+
(
GA4)L2
+ eL 3
GA5
This does not seem to be feasible.
etc.
A 3.
Instead, letting
tend
to zero leads to
8L
8w
L2
2
GA5 A3
A--I
(5.)
GA--4
This will be used to define L.
Let
-
be a new variable and consider
XA5
A
L2=_
3
(5.2)
Note that the right side of (5.2) is analytic where w # 0 and
I
A4 # 0. So let L
L(Wl,W2,t,e, ) Pl(W2) + P2(Wl,W2,t,e,) be an
analytic solution on (5.2) and assume that none of the expressions A
vanish when L
c
o
Pl(w2 )"
Now since the value of
2 ((Wl,W2,t,e,G(wl,t))
w
-2(’(Wl,W2,t,e,)t)
we see that
solution of (5.1) for any G.
is analytic.
$1’
Let K
G
s
for )t
L(wl,w 2,t,e)
G(Wl,t) is the same as
L(wl,w 2,t,,G(w 1,t)) is a
I
Moreover L e C since G is continuous and L
(L,Wl,W2,t)
and L
L.
We now prove the solvability near suitable points of the Cauchy-Kowalewski
system.
The variable
gives our functions the universal character referred
to previously.
LEMMA I.
Let (a,b,c) be such that
Sl(Pl(b)a,b,c)
the Jacobian of (3.1) is nonzero at (a,b,c,e).
# 0.
Then, for small t,
263
NONLINEAR DIFFERENTIAL EQUATIONS
If the Jacobian of (3.1)
PROOF.
-A2LI + A4L2
0, then
+ A5
0
The subsidiary equations of (5.3) are:
dwl dw2
A
-A2
dL
so that
-A
5
4
Thus
AIA5 A3A4
-As
A--4
GAs A3
dL
But from (5.1),
dL
dw
2
0.
w
But
AIA5 A3A4
1
(Wlao4 WlW2ao2 + w2s o)(--o +
z).
So
w
L
E+0
(WlUo4
1
WlW2ao2 + w2uo) (L---o + z)
(WlUo4
WlW2Uo2 +
0.
However
w
L
e-0
I
W2Uo) (--o +
Sl(Pl(W2)’Wl’W2’t)
z)
# 0 and the
proof is complete.
We next consider continuity in order to apply the implicit function
theorem to (3.1).
We first observe that
Now consider the case where
LEMMA II.
PROOF.
L A # 0.
1
L A # 0, but
4
If
Pl(W2)
L(Wl’W2’t’G(Wl’t))
0, then
4)
GA
L
There is at most one function G such that
L(Wl’W2’t’e)
L A4
+
L (
-I
0.
GA4)
eP2(Wl’W2’t’e’G(Wl’t))
So it and its partials with respect to
Wl,W2,t
do not contain G as 0.
n
+
I ’w2’ t) +
Z cn (L,w ’w2’ t)
I
n=0
c
o(L ’Wl’W2’ t)
Cl(L
w
0.
c2(L,Wl,W2
Since
t)e 2 +
L. K. WILLIAMS
264
the same holds for it.
eAl
0A4
G
So
L A are independent of G.
eO 4
L A and
e0 I
Thus
,
This completes the proof.
In the sequel, we ignore this possible exception and assume that
L
GA--4)
(A--I
LEMMA III.
0 for any G.
S2(Pl(b),a,b,c)
If (a,b,c) is such that
e > 0 such that the left sides of (3.1) are C I at
PROOF.
of
$2,
Based on analytic properties of
0, there is an
(a,b,c,e).
and the nonvanishing
VI,V2,L,KG
we will not give details.
Choosing constant values for
Wl,W2
in (3.1), we can get
Cl(e),c2(e)
so
that left sides vanishes and apply the implicit function theorem to (3.1).
Then we solve for
Wl(t,e)
and
w2(t,e).
Here
Cl,C 2 come
from equation (2.1)
of section 2.
6.
THE PRINCIPAL DIFFERENTIAL
EUATION.
We now consider the differential equation
dx
DEFINITION
Y
(6.1)
g(x,y)
L wl(t,e).
W
l(t)
It will be shown that
Wl(t)
Of course we change y,x
satisfies (6.1).
to W" t respectively.
1,
We begin this process wlth
THEOREM II.
PROOF.
and also
L
/
Let S
Pl(W2)
0 as e
/
d
l,W--2,t--). Then wl(t
P2(Wl,W2,t,G(Wl,t)) so that
0 at
1
+
O.
Thus
L1,L 3
/
0 as
-
)
--i
0.
/
0
Gl,t-6)
as
as
+ 0
/
0.
NONLINEAR DIFFERENTIAL EQUATIONS
Thus
R
AIA5 A3A4
+ A5
A4L2
Also
A
GA
I
eL
3
+
G(A4L 2
+
4
AIL 2 + A 3
L +
3
-eL +
1
+ A5)
(A1
+ A5
A4L2
+ GS 1
GA4)L 3
+
+
GS
Therefore
GA
A
I
AIL 2 + A2L 3 + A3
i
SI
4
-A2L1 + A4L2 + A5
So
i S--1
as
/
By the last theorem,
->
L
e->0
G(Wl (t), t)
265
0 and S
d
0.
1 #
This completes the proof.
G(wl(t,e),t)
wl(t’)
G( L wl(t,e),t)
But also it is true [2: P.461] that
L
(6.2)
w
2 t)
y’
Let J (Wl, t)
L(wI w2 t e,J(wI t)) and
Let Q be the set of points in
(i)
W
W
7 (a) PARTICULAR SOLUTIONS.
L* (wI
l’(t).
Wl(t,e))
G(Wl(t),t)
I (t)
PARTICITLAR AND GENERAL SOLUTIONS OF
So
7.
Wl(t,e)
G(x,y).
I
C
* (wl,w2,t)
(Wl,W2,t)-space
n
o(L*,wI ,w2
where
0
(3) S 1 # 0
(2) Co # 0
(4) S 2 # O.
I #
Let Q be the projection of Q on the (Wl,t) plane.
w
The Universal Cauchy-Kowalewski System
DEFINITION.
(Wl’W2’ t’ e’ )
o (L’Wl’W2, t)
DEFINITION.
F
+
1
w
I
eVl(W1
tw2,cl,C2).
t)
L.K. WILLIAMS
266
DEFINITION.
F
DEFINITION.
F
2
3
=_ w
L(Wl,W2,t,e,% -V2(wI
2
=- AI J(wl,t)A4 with
F
DEFINITION.
F
The system
+
tw2,cl,c2).
% replaced by
J(wl,t).
0
I
is also called the Universal
0
2
F#0
Cauchy-Kowalewski System.
We refer to it in the following
THEOREM III.
P of
i
(i)
Let P e Q.
J(wl’t)
In FI,F
2
There is a region in which the solution through
is determined as follows:
J(Wl,t
replace % by
and
(2)
Equate the results in (2.1) to zero.
(3)
Solve the resulting system for
(4)
Take the limit of
PROOF.
Let P
such that
wl(t,
(a,to)’
*(a,b,to E)
P
wl(t,)
Q"
Since
Let (V
+
tob)
2(a + tob)
and
.
w2(t,).
as e + 0.
L,V-2)
# 0.
Vl(a
E
by suitable functions of
Cl,C 2
# 0, there
Co(Pl (b)’a’b’to)
is an
be a solution of (2.1) such that
*(a b,t o ,)
b2
L*(a,b,to,e).i
Solve the syst era:
to get suitable c
Since S
I
(1)
Vl(a
+
tob’Cl’C2)
e(a
a
b t
(2)
V2(a
+
tob,CI,c2)
b2
L*(a,b,to,)
Cl(),
c
2
+
0
0
c2( ).
0 our system has nonzero Jacobian.
I #
get the result.
o’ e)
l(t,e)
We solve for w
and
267
NONLINEAR DIFFERENTIAL EQUATIONS
7(b) GENERAL SOLUTIONS.
Alternatively, eliminating w
2
from the Universal
Cauchy-Kowalewski System we get
X(Wl,t,e,,Cl,C2)
where
cl,c 2
(7.1)
0
are constants.
The general solution of a specific equation is obtained as follows:
(I)
Replace
(2)
Take the limit as e + 0 of the result.
by
G(Wl,t )
in (7.1).
X is derived from L and K and is like the familiar differential quotient in
generality.
REFERENCES
i.
Garabedian, P. R.
2.
Olmsted, J.
Partial Differential Equations
Advanced Calculus, Appleton-Century.
John Wiley, 1964.