Ordinary Differential Equations Of The Third

Order: Damped Harmonic Motion

A Little Humor/Philosophy Before We Begin(?)

If a Sine Wave describes uniform deviations from the norm

then the Cosine function has a similar form

but with an appropriate shift in its phase.

But are not these sine wave deviations the continuous oscillations

that induce the periodic vibrations

that are characteristic of natures’ “sinning” ways?

If no natural phenomena could be described by a sin function

at any time, intersection or junction

then would the world be without the acts of sinning?

Would there have been deviations from the mean

in all that has been heard or seen

up until now, starting from the beginning?

But if there were no longer deviations from the mean

no differences would remain

and as for the “meaning of the mean”

there would be no need to explain!

Let us now consider the general homogeneous second-order ordinary differential equation

y^’’ + By^’ + Cy = 0 (eq.1)

where the coefficients B and C are constant. When B = 0

and C > 0 the solutions of (eq.1) are pure “un-damped”

sine or cosine functions. In particular if B = 0 and

C = 1, then (eq.1) reduces to

y^’’ + y = 0 (eq.2)

As is well known, two (linearly) independent solutions

(eq.2) are cos(x) and sin(x). These functions are as-

sociated with many identities such as

cos²(x) + sin²(x) = 1. (eq.3)

They also appear in the fundamental identity

e^ix = cos(x) + i*sin(x) (eq.4)

Equations (eq.3) and (eq.4), (known as Eulers’

Formula), play indispensable roles in math, science,

engineering and theoretical physics. The reader will

recall that since e^ix is a linear combination of cos(x)

and sin(x), it is also a solution of the ordinary

differential equation (O.D.E), (eq.2).

Let us pose the question: are there such identities

associated with 3^rd order O.D.E.’s with constant coeffi-

cients? Well why don’t we create a 3^rd order O.D.E by

differentiating (eq.1) and see what we can find out?

Note: Later when we discuss the kinematics of simple

and damped harmonic motion we will replace the

independent variable x by t (which will denote time).

It is convenient to write (eq.1) in the following

form

y^’’ = - By^’ - Cy (eq.5)

Differentiating (eq.5), recalling that B and C are

constants, with respect to x results in

y^’’’ = - By^’’ – Cy^’ (eq.6)

Now substitute the expression for y^’’ , from (eq.5)

into (eq.6) and simplify the results. We obtain

y^’’’ + (C – B²)y^’ – BCy = 0.

If we choose units so that

C = B²

then the equation above reduces to

y^’’’ – B³y = 0. (eq.7)

Before continuing, let’s note that with B = 0 and

C = k², we may write (eq.1) as

y^’’ + k²y = 0 (eq.8)

Equation (eq.8) may be referred to as the one

spatial dimensional “undamped wave” equation.

When B ≠ 0 then (eq.1) may be thought of as the

one spatial dimensional damped wave equation.

Equation (eq.7)is a (3^rd order) generalization of

(eq.1) in which, for convenience, we have chosen

units so that C = B² .

The reader is probably familiar with many cases

in (math) physics where the units of physical quantities

and the units associated with constants of proportionality

have been chosen for the purpose of simplifying equations.

For example, in quantum field theory it is often the case

that units are chosen so that both c the speed of light and Plancks’ constant are set equal to 1 with no loss of

generality.

If we write y⁽ⁿ⁾ to denote the nth derivative of y with

respect to x, then both (eq.7) and (eq.8) may be derived

from

y⁽ⁿ⁾ + Lⁿy = 0 (eq.9)

by letting n = 3 and L = -B in the case of (eq.7)

and n = 2 and L = k (or –k) in the case of (eq.8).

The implications of (eq.9), when n is a non-negative

integer different from 2 or 3, will be discussed

later.

In what follows we are motivated by the possi-

bility of finding identities, associated with (eq.7),

that correspond to the identities that are derived

from (eq.2).

Before continuing, we will need to discuss the

preliminaries of the theory of Cubic Complex

Variables. The algebra (algebraic ring) of complex

numbers has been generalized to the nth order case

by the present author. The 2^nd order case is just

ordinary complex numbers and the 3^rd order case has

been referred to as Cubic Complex numbers or

variables. It is well known, of course, that every

algebraic field is also an algebraic ring but that

the converse is not necessarily true.

Recall that the complex “imaginary” unit i may

be defined by

i = (-1)^(1/2)

But the multiplicative identity 1 and i may be put in a

one to one correspondence with unit vectors in the plane

as follows:

1 ↔ (1,0)

i ↔ (0,1)

The present author was motivated to write, in the early

1980’s, a more abstract definition of i.

i = (-1,0)^(1/2)

so that

i² = (-1,0) = -1

The reader will recall from elementary abstract algebra

that i is the generator of cyclical group of order 4. We

may write

i¹ = i

i² = -1

i³ = -i

i⁴ = 1

Making use of the fact that i² = -1, as the reader knows

well, we may form the product of two arbitrary complex

variables u = x + iy and v = a + ib,

uv = vu = ax –by + i(ay + bx).

Using the ordered pair notations (corresponding to vectors

in the plane), we may also write

(x, y)(a, b) = (ax – by, ay + bx)

Now we will turn our attention to Cubic Complex Variables.

We will refer you elsewhere for a more thorough discussion

of Cubic Complex Numbers (Variables) link . But presently we will survey the preliminaries. We will start by denoting the set of Cubic Complex Numbers by C3. The set C3 will be defined as the set of ordered triples of ordinary complex numbers C2. This

set has been endowed with the operations of vector addition and

a commutative rule for multiplication.

In actuality C3 has been shown to be a ring with C2 as a

sub-ring. The reader will recall that C2 is also of course a field. But all fields are also rings (even though the converse is not necessarily true). Clearly C3 contains the set of 3D vectors (with real components) as a subset. We will make the

identification.

1 <=> (1,0,0).

We now define the entity j as follows:

j = (-1,0,0)^(1/3) (eq.10a)

For brevity we write¹

j = (-1)^(1/3) (eq.10b)

and

j^3 = -1 (eq.11)

(The reader will note that j as defined above is different from

any of the three cube roots of -1; where, in that instance, we are making the identification) -1 <=> (-1,0). The three

ordinary cube roots of -1 are -1 and a pair of complex

conjugate numbers.)

It readily follows that

j^3 = -1 (eq.12a)

j^4 = -j (eq.12b)

j^5 = -j^2 (eq.12c)

j^6 = 1 (eq.12d)

An arbitrary cubic complex variable u may be written as

u = a + jb + (j^2)c (eq.12e)

where, in general, a, b and c are (ordinary) complex numbers. If

v = x + jy + (j^2)z (eq.12f)

is another arbitrary cubic complex variable

then we may write the sum and product as

u + v  =  a + x + j(b + y) + (j^2)(c + z) (eq.13)

uv = ax - bz- cy + j(ay + bx - cz) + j^2(az + cx + by)  (eq.14)

The operation of addition is obviously commutative and

it is not difficult to show that the product is also

commutative.

The sum and product operations may also be expressed in

terms of ordered triples as follows:

(a, b, c) + (x, y, z) = (a + x, b + y, c + z)

(a,b,c)(x,y,z)  = (ax – bz - cy, ay + bx - cz, az + cx + by)

By definition, if u = a + jb + (j^2)c is a cubic complex

number and if v = x + jy + (j^2)z is also a cubic complex

number, then

u = v if and only if

a = x (eq.14a)

b = y (eq.14b)

c = z (eq.14c)

We will now make use of the cyclicality (of order six) of

j^n, (where n is an integer) to define the Cubic Complex

Exponential e^(jx). Later on we will examine the more General

form of the Cubic Complex Exponential: e^{jx + (j^2)y}.

It will be seen that the Cubic Complex Exponential e^(jx) will generate solutions of the third order O.D.E.

y^’’’ – B³y = 0,

just as the complex exponential e^ix generates solutions of the

second order O.D.E.

y^’’ + k²y = 0

We have

e^jx = 1 + jx + (1/2)(jx)^2 + (1/6)(jx)^3 + ... + (1/n!)(jx)^n

+ ...

e^jx = 1 + jx + (1/2)(j^2)(x^2) + (1/6)(j^3)(x^3) + ... +

(1/n!)(j^n)(x^n) + ... (eq.15)

Now since j^n will always equal (+/-)1, (+/-)j or (+/-)j^2,

where the symbol +/- denotes “plus or minus”, the above

equation may be written as

e^jx = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ...+ j[x-(1/4!)(x^4)

+(1/7!)(x^7)+...]+j^2[(1/2!)(x^2)-(1/5!)(x^5)

+(1/8!)(x^8)+ ...] (eq.16)

It will be noted that three separate infinite series

are indicated on the right of (eq.16). We may define

them, using the notation F₁, F₂ and F₃, as follows:

F₁ = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ... (eq.17a)

F₂ = x - (1/4!)(x^4) + (1/7!)(x^7) +... (eq.17b)

F₃ = (1/2!)(x^2)-(1/5!)(x^5) +(1/8!)(x^8) + ... (eq.17c)

The reader is reminded that even though the (conventional) notation + ... appears after the third term, on the

right sides of each of the three equations directly above,

in actuality the terms will alternate in sign. The

convergence of these infinite series can be established

without difficulty.

So then with the aid of equations 17a, 17b and 17c, we may

re-write (eq.16) simply as

e^jx = F₁ + jF₂ + (j^2)F₃ (eq.18a)

or equivalently as

e^jx = F₁ + jF₂ + j²F₃ (eq.18b)

We will refer to (eq.18a or b) as the Cubic Complex

(versions of the) Euler Formula. These are actually

identities because in each equation, by the definitions

set forth, each side is merely an equivalent

representation of the other side.

For His marvelously magnificent

creative ways

lets stop and give the Creator

some praise.

He is too often taken completely

for granted

on this tiny little planet!

Yet some of us have the audacity

to speculate about the beginnings

of the Universe

as if we could accurately circum-navigate

space by moving through time in reverse!

Using summation notation we may also write (eq.18b) as

e^jx = ^kF_k+1

The three functions F₁, F₂ and F₃ will be called the fundamental

functions of Cubic Complex Variable (CCV) theory. The first such

function F₁ will be called the principal function of CCV and it

will be seen to play a role similar to the cosine function in

ordinary complex variable theory.

We will now show that the three fundamental functions are

independent solutions of

y^’’’ + y = 0. (eq.19)

which may be obtained from (eq.7) by letting B = -1.

Recall from (eq.17a) that

F₁ = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ...

differentiating (term by term) with respect to x gives us

F₁^’ = -[(1/2!)(x^2)-(1/5!)(x^5) +(1/8!)(x^8) + ...]

where the symbol ( ^’ ) denotes differentiation (with respect

to x). But the expression inside of the brackets, in the

above equation, is just F₃ as defined by (eq.17c). Therefore

we may re-write the equation directly above as

F₁^’ = - F₃ (eq.20).

Differentiation of both sides of (eq.20) results in

F₁^’’ = - F₃´ (eq.21)

To determine F₃´ we differentiate the right side of

(eq.17c). We obtain

F₃´ = x - (1/4!)(x^4) + (1/7!)(x^7) +...

It is seen that, according to (eq.17b), the right of the

the above equation is F₂. So we have

F₃´= F₂ (eq.22)

From equas.(21) and (22) it immediately follows that

F₁^’’ = - F₂ (eq.23)

Differentiation of both sides of (eq.23) with

respect to x results in

F₁^’’’ = - F₂^’ (eq.24)

Differentiation, with respect to x, of the

defining equation for F₂, (eq.17b),

F₂´ = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ...

But the infinite series on the right side of the

equation above is by definition F₁ . So we have

F₂´ = F₁ (eq.25)

Substitution of this result into (eq.24) results in

F₁^’’’ = - F₁

or equivalently

F₁^’’’ + F₁ = 0 (eq.26)

Now let y = F₁ so that

y^’’’ + y = 0. (eq.27)

Therefore y = F₁ is a particular solution of

(eq.27). In a similar fashion, it can also be

shown that F₂ and F₃ are also independent

solutions of (eq.27). It will also be noted that

(eq.27) may be obtained from (eq.7) by setting

B = -1.

So then we have shown that the components of the

Cubic Complex exponential

e^jx = F₁ + jF₂ + j²F₃

are independent solutions of the third order O.D.E.

y^’’’ + y = 0

just as the ordinary complex exponential

e^ix = cos(x) + i*sin(x)

has components that are solutions of the second order O.D.E.

y^’’ + y = 0.

It is hereby conjectured that these results may be generalized to the nth order case involving

y⁽ⁿ⁾ + y = 0

The double basses stood on the floor

of the great harmonic foundation

allowing other voice to soar

with higher frequencies of vibration.

While the cellos played an octave higher

with a more interesting part

and as the tenor of the string choir

its melodies often touched the heart.

The violas are the altos

with a husky sweet tone

and although not as sweet as the violins

they seldom if ever drone.

And then there is the “Violino Principali”.

which can whisper to the wind

or to the depths of the heart

and the strings that lie within.

But can the strings of the choir

help to unite the sciences and the arts

in the region where a supposition of cosine

waves begins

and where the melodic line starts.

Now we will show that the three fundamental functions, F₁ ,F₂ and F₃ can be obtained, using Taylor Series, as solutions for initial value problems. First we will find a particular solution

of

y^’’’ + y = 0 (eq.28a)

such that the following conditions are satisfied:

y(0) = 1, (eq.28b)

y^’(0)= 0 (eq.28c)

y^’’(0) = 0 (eq.28d)

Now let x = 0 in (eq.28a) and obtain

y^’’’(0) + y(0)= 0

but y(0)= 1, from (eq.28b) so we have

y^’’’(0) + 1 = 0 or

y^’’’(0) = -1 (eq.29)

If we now differentiate (eq.28a) with respect to x we

will obtain

y⁽⁴⁾ + y^’ = 0 (eq.30)

where the notation y⁽⁴⁾ is employed to denoted the 4^th

derivative of y with respect to x. In general, we will

denote the nth derivative of y with respect to x by

the symbolism y⁽ⁿ⁾ for (positive) integer n ( > 3 ).

By setting x = 0 in (eq.30) we arrive at the following

result

y⁽⁴⁾(0) + y^’(0)= 0

but from (eq.28c) we have

y^’(0)= 0, therefore we may write y⁽⁴⁾(0) + 0 = 0 or

y⁽⁴⁾(0) = 0 (eq.31)

Successive differentiation of (eq.30) will give us

y⁽⁵⁾ + y^’’ = 0 (eq.32)

and

y⁽⁶⁾ + y^’’’ = 0 (eq.33)

Setting x = 0 in (eq.32) and then substituting

y^’’(0) = 0, from (eq.28d), will leave us, after

simplifying, with

y⁽⁵⁾(0) = 0 (eq.34)

And finally, in this sequence of calculations,

if we set x = 0 in (eq.33) and then substitute

y^’’’(0) = -1 from (eq.29), we arrive at the

following

y⁽⁶⁾(0) = 1 (eq.35)

Now, thanks to the information provided by

equas.(28b,28c,28d,29,31,34 and 35), we have the

values of y and its first six derivatives evaluated

at x = 0. Therefore we can write the first seven

term of a Taylor Series² expansion of the function

y = y(x) near x = 0. We obtain

y = y(0) + y^’(0)x + (1/2)y^’’(0)x² + (1/6)y^’’’(0)x³

+(1/24)y⁽⁴⁾(0)x⁴ + (1/5!)y⁽⁵⁾(0)x⁵

+ (1/6!)y⁽⁶⁾(0)x⁶ + ...

Now making the appropriate substitutions and dropping the

zero terms (of course) we arrive at

y = 1 – (1/3!)x³ + (1/6!)x⁶ + ... (eq.36)

Except for the differences in the notation used for

exponents, it is seen that the rights of (eq.36) and

of (eq.17a) are identical. Accordingly, they must

represent the same (convergent) infinite series and so

we can state that y = F₁, as defined by (eq.17a), satisfies

(eq.28a). It can be easily verified that y satisfies

the auxiliary conditions of equas.(28b, c and d).

Following a procedure, similar to that used above, it

can be shown that y = F₂ satisfies the equation

y^’’’ + y = 0

such that the following conditions are satisfied:

y(0) = 0, y^’(0)= 1 and y^’’(0) = 0.

It can also be demonstrated that y = F₃ satisfies the same equation subject to the conditions

y(0) = 0, y^’(0) = 0 and y^’’(0) = 1.

Let cos(x) and sin(x), which are two well known linearly

independent solutions of (eq.2), be denoted by f₁ = cos(x)

and f₂ = sin(x). We then have

f₁^’ = df₁/dx = - f₂ = - sin(x).

So then the famous identity

cos²(x) + sin²(x) = 1 may be written as

(f₁)² + (f₁^’)² = 1 (eq.37a)

It can also be easily shown that

(f₂)² + (f₂^’)² = 1 (eq.37b)

We will now derive the corresponding identities

for the third order case which involve concepts

from CCV theory. But first it will be helpful if

we recall some familiar relationships from ordinary

complex variables.

Let

X + iY = (a + ib)(x + iy)

then it is easy show that

X = ax – by (eq.38a)

Y = bx + ay. (eq.38b)

If we let a and b be constants and then consider

equas.(38a & b) as a system of linear equations in

x and y, then the matrix of coefficients of that

system may be denoted by M₁ where M₁ is given by

M₁ = a -b (eq.39)

b a

where for typographical convenience we have omitted

the parenthesis that are conventionally used to

enclose the elements of matrices. We have,

writing the determinant of M₁,

det(M₁) = a² + b² (eq.40)

Now the reader will recall, or can easily verify,

that

X² + Y² = (a² + b²)(x² + y²) (eq.41)

where X and Y are defined as in equas.(38a & b).

It will also be noted that the left side and both

factors on the right side of (eq.41) are (2^nd

degree) expressions that have the same algebraic

form as does the right side of (eq.40). Also if

a = cos(θ) and b = sin(θ), then of course

a² + b² = 1

and equas.(38a & b) represent an orthogonal coordinate

transformation.

Now for the CCV counterparts of the foregoing, let u

v be defined as in equas.(12e &f) above. Then the

product of u and v is given by

uv = ax - bz- cy + j(ay +bx -cz) + j^2(az+cx+by) (eq.42a) 

on account of (eq.14).

Since u and v are cubic complex variables, their

product is one also. Let the product uv be, the

cubic complex variable, given by

uv = X + jY + j²Z

then (eq.42a) may be written as

X + jY + j²Z = ax - bz- cy + j(ay +bx -cz) + j^2(az+cx+by)

(eq.42b)

Then therefore, by the definition of the equality

of cubic complex numbers, see equas.(14a,b and c)

we have

X = ax - bz- cy

Y = ay + bx - cz

Z = az + cx + by

It is convenient to rearrange the order of the terms

on the right sides of the three above equations as

follows:

X = ax – cy - bz (eq.43a)

Y = bx + ay - cz (eq.43b)

Z = cx + by + az (eq.43c)

Now if we consider equas.(43a,b & c) be to be a

system of three linear equations in x, y and z, then

the matrix of coefficients, which we denote by M₂ is

given by

M₂ = a -c -b

b a -c

c b a

the determinant of the matrix M₂ is given by

det(M₂) = a³ – b³ + c³ + 3abc (eq.44)

After a somewhat tedious calculation, it can be

shown that

X³ - Y³ + Z³+ 3XYZ =

(a³ – b³ + c³ + 3abc)(x³ – y³ + z³ + 3xyz) (eq.45)

where X,Y and Z are defined by equas.(43a,b & c).

So then (eq.45) is the CCV theory analog to

(eq.41) which is a fundamental in ordinary complex

theory. Recall, for example, that the square of

the absolute value of the complex number x + iy

is x² + y² .

It can be shown, in a straight forward fashion,

that

(F₁)³ – (F₂)³ + (F₃)³+ 3(F₁)(F₂)(F₃) = 1 (eq.46)

where F₁, F₂ and F₃ , each a function of x, are defined by

equas.(17a,b and c). The relationship holds true for all

real x. So (eq.46) corresponds to the identity

of (eq.3) which is associated with complex

variable theory.

Kinematics Of Simple Harmonic Motion And Damped

Harmonic Motion

With a convenient choice of units, associated with the

relevant physical constants, simple harmonic motion may

be described by the equation

x^’’ + x = 0. (eq.47)

Where x = x(t) and (in this section)

the primes indicate differentiation

with respect to t.

A particular solution of this equation is

x = cos(t), so that x^’ = -sin(t), therefore

(x)² + (x^’)² = 1 (eq.48)

For now we will merely make note of the well known fact

that (eq.48) is an identity that involves the position

x = x(t) and the velocity x^’ of an object that is in

simple harmonic motion.

Let’s also make note of the fact that conventionally

the arguments of the sine and cosine functions, in

connection with simple harmonic motion, are often

written as ωt + β where ω is the angular frequency

and β is a dimensionless constant (initial angular

displacement). For convenience we are choosing

units and initial conditions such that ω = 1

and β = zero. This should not, in anyway,

affect the generality of the principles

discussed and proposed.

The reader will recall that previously

we let y = y(x) and y^’ denote dy/dx, etc.,

and demonstrated how the second order

equation

y^’’ + By^’ + Cy = 0

could be transformed into the third

order equation

y^’’’ + y = 0.

Now with a simple change in notation and variables, the

same results apply to

x^’’ + Bx^’ + Cx = 0 (eq.49)

which can transformed into

x^’’’ + x = 0. (eq.50)

where now x = x(t) and x^’ = dx/dt, etc.,

The second order Equation (eq.49) may be

interpreted as representing damped harmonic motion.

However, a relationship of form of (eq.48) does

not identically hold true for the solution (and

its 1^st order derivative) of the 2^nd order

equation for damped harmonic motion. But,

alas, when (eq.49) is transformed into the

form of (eq.50), such a relationship does

indeed exist.

For notational convenience replace x by t in

equas.(17a,b and c) and obtain

F₁ = 1 – (1/3!)(t^3) + (1/6!)(t^6) + ... (eq.51a)

F₂ = t - (1/4!)(t^4) + (1/7!)(t^7) +... (eq.51b)

F₃ = (1/2!)(t^2)-(1/5!)(t^5) +(1/8!)(t^8) + ... (eq.51c)

where F₁, F₂ and F₃ are now denoting functions of

t.

Now let

x = F₁. (eq.52a)

Therefore

x^’ = dx/dt = -F₃, (eq.52b)

x^’’= -F₂ (eq.52c)

x^’’’ = - F₁ (eq.52d)

We now can write

(x^’’)³ – (x^’)³ + x³ + 3(x^’’)(x^’)x =

(-F₂ )³ – (-F₃)³ + (F₁)³ + 3(-F₂)(-F₃)F₁ =

-(F₂ )³ + (F₃)³ + (F₁)³ + 3(F₂)(F₃)F₁ =

(F₁)³ - (F₃)³ + (F₃)³ + 3(F₁)(F₂)F₃ = 1. (eq.53)

The last member of the above extended equation holds true

as a consequence of (eq.46). Re-writing (eq.53) using

only the first and the last members

(x^’’)³ – (x^’)³ + x³ + 3(x^’’)(x^’)x = 1 (eq.54)

We can now state, that for an object executing

damped simple harmonic motion, that (eq.54) holds

identically true: when convenient choices of the

initial conditions and the units of the

constants of proportionality are adopted.

The instantaneous acceleration, velocity and

position are given by x^’’, x^’ and x

respectively.

It is strikingly noteworthy to observe that (x^’)³,

the cube of x^’, which is also associated with the

dissipative force of friction, [in (eq.49) the

O.D.E. governing damped simple harmonic motion],

is preceded by a negative sign in (eq.54). This

fact adds weight to the assertion that (eq.54) is

the damped simple harmonic motion counterpart

of (eq.48).

While more details of the present theory will be presented

later, it should be noted that these concepts can be generalized to higher orders and degrees.

*****

And generalizations to higher orders and degrees affords mankind a set of potential keys to unveil all of the uncertainties that are at the core of our limitation. For He who is the Father of all creation has endowed us with the power of imagination with which we can imagine that we are all made in his image as

it so has been written as inspired by He from whom nothing is

hidden.

And so He who knows all the stars by name deserves all of His

Universal fame. For He initiated their evolutions and directed

them into the institutions of the wonders that they have

became.

Note:

Sometimes when a poet does theorize on philosophical tracts

he has permission to occasionally revise grammatical syntax.

Footnotes:

1.)

The author has also generalized the concept of the cubic

complex unit to the nth order. The nth order complex unit

j_n been defined by

j_n = (-1,0,0,...)^(1/n)

where (1,0,0,...) is an n dimensional unit vector.

2.)

The author outlined in an informally distributed 1980 paper, a generalization of the method used above for finding Taylor Series solutions of nth order ordinary differential equations: subjected to a set of n auxiliary conditions.

Your reading of this preliminary draft is highly appreciated.

Future postings will present further developments, more

extensive applications and any needed clarifications.

This discussion has been presented under the heading of mathematical “phiction” as a matter of choice. But it must be remembered that much of the science fiction of yesterday is responsible for many of the engineering wonders of today. Let’s hope that an effort will be made for a more rapid review of worthy science innovations and/or math phiction formulations

in order to more quickly determine the merits of potentially productive conceptualizations.