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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

 Isaac Newton 
Newton, 1666


Leibniz, 1675

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"The Rise of Calculus", in the History of Mathematics from St. Andrews.
Contemporaries of Descartes, Fermat, Pascal & Huygens  by  Rouse Ball  (1908)

Numerical integration and the Newton-Cortes formulas.
calc101.com:  Automated Calculus (Derivatives, Integrals, Polynomials, etc.)
S.O.S. MathCalculus & Differential Equations.
The Birth of Calculus: 1666 & 1675   Jeremy Gray  (1986.  Aired 1992-07-27).


 f '    =    f
AlisonWonder (2002-06-23)   The Basics:
What is a derivative?

Let's give the traditional approach first.  This will be complemented by an abstract glimpse of the  bigger picture,  which is more closely related to the way people actually  use  derivatives, once they are familiar with them.

For a given real-valued function f of a real variable, consider the slope (m) of its graph at some point.  That is to say, some straight line of equation y = mx+b (for some irrelevant constant  b) is tangent to the graph of f at that point.  In some definite sense, mx+b is the best linear approximation to f(x) when x is close to the point under consideration...

The tangent line at point x may be defined as the limit of a secant line intersecting a curve at point x and point x+h, when h tends to 0.  When the curve is the graph of f, the slope of such a secant is equal to [ f(x+h)-f(x) ] / h, and the derivative (m) at point x is therefore the limit of that quantity, as h tends to 0.

The above limit may or may not exist, so the derivative of f at point x may or may not be defined.  We'll skip that discussion.  The popular trivia question concerning the choice of the letter "m" to denote the slope of a straight line (in most US textbooks) is discussed elsewhere.
      Way beyond this introductory scope, we would remark that the quantity we called h is of a vectorial nature (think of a function of several variables), so the derivative at point x is in fact a tensor whose components are called partial derivatives.
      Also beyond the scope of this article are functions of a complex variable, in which case the above quantity h is simply a complex number, and the above division by h remains thus purely numerical (albeit complex).  However, a complex number h (a point on the plane) may approach zero in a variety of ways that are unknown in the realm of real numbers (points on the line). This happens to severely restrict the class of functions for which the above limit exists.  Actually, the only functions of a complex variable which have a derivative are the so-called analytic functions [essentially: the convergent sums of power series].

The above is the usual way the concept of derivative is often introduced. This traditional presentation may be quite a hurdle to overcome, when given to someone who may not yet be thoroughly familar with functions  and/or limits.  Having defined the derivative of f at point x, we define the derivative function g = f ' = D( f ) of the function f, as the function g whose value g(x) at point x is the derivative of f at point x.

We could then prove, one by one, the algebraic rules listed in the first lines of the following table.  These simple rules allow most derivatives to be easily computed from the derivatives of just a few elementary functions, like those tabulated below  (the above theoretical definition is thus rarely used in practice):

and  v  are functions of  x,  whereas  a, b and  n  are constants.
  Function   f  Derivative   D( f ) = f ' 
Linearitya u   +   b v a u'   +   b v'
Product Rule
(Leibniz' Law)
u ´ v u' ´ v   +   u ´ v'
u / v [ u' ´ v  -  u ´ v' ]  /  v 2
Chain Ruleu(v) v' ´ u'(v)
Inversionv = u-1 1 / u'(v)
Constantsa 0
Powersxn n x n-1
Logarithmsln |x| 1/x   =   x -1
ln |ln |x| | 1 / (x ln |x|)
ln | ln |ln |x| | | 1 / ( x ln |x| ln |ln |x| | )
Exponentialse xe x
a xln(a)  a x
sin xcos x
cos x- sin x
tg x1 + (tg x)2
ln | cos x |- tg x
sh xch x
ch xsh x
th x1 - (th x)2
ln ( ch x )th x
Inverses of
arcsin x    (for |x|<1)   1 / Ö(1-x2 )
  arccos x = p/2 - arcsin x  -1 / Ö(1-x2 )
arctg x  1 / (1 + x2 )
Inverses of
argsh x   1 / Ö(1+x2 )
argch x   (for |x|>1)  1 / Ö(x2 -1)
argth x   (for |x|<1)   1 / (1 - x2 )
argth 1/x   (for |x|>1)
gd x   =   2 arctg ex - p/2  1 / ch x
 gd-1 x  =   ln tg (x/2 + p/4)   1 / cos x

One abstract approach to the derivative concept would be to bypass (at first) the relevance to slopes, and study the properties of some derivative operator D, in a linear space of abstract functions endowed with an internal product (´), where D is only known to satisfy the following two axioms (which we may call linearity and Leibniz' law, as in the above table):

D(au + bv)= a D(u)  +  b D(v)
D( u ´ v )= D(u) ´ v  +  u ´ D(v)

For example, the product rule imposes that D(1) is zero [in the argument of D, we do not distinguish between a function and its value at point x, so that "1" denotes the function whose value is the number 1 at any point x].  The linearity then imposes that D(a) is zero, for any constant a.  Repeated applications of the product rule give the derivative of x raised to the power of any integer, so we obtain (by linearity) the correct derivative for any polynomial.  (The two rules may also be used to prove the chain rule for polynomials.)

A function that has a derivative at point x (defined as a limit) also has arbitrarily close polynomial approximations about x.  We could use this fact to show that both definitions of the D operator coincide, whenever both are valid (if we only assume D to be continuous, in a sense which we won't make more precise here).

This abstract approach is mostly for educational purposes at the elementary level.  For theoretical purposes (at the research level) the abstract viewpoint which has proven to be the most fruitful is totally different:  In the Theory of Distributions, a pointwise product like the above (´) is not even defined, whereas everything revolves around the so-called convolution product (*), which has the following strange property concerning the operator D:

D( u * v )   =   D(u) * v   =   u * D(v)

To differentiate a convolution product (u*v), differentiate either factor!

 Isaac Barrow 
 (1630-1677) (2002-06-26)
What's the "Fundamental Theorem of Calculus" ?

Once known as Barrow's rule, it states that, if  f  is the derivative of F, then:

F(b) - F(a)   =    ó
  f (x)  dx

In this, if  f  and F are real-valued functions of a real variable, the right-hand side represents the area between the curve y = f (x) and the x-axis (y = 0), counting positively what's above the axis and negatively  [negative area!]  what's below it.

Any function  F  whose derivative is equal to  f  is called a  primitive  of  f  (all such primitives simply differ by an arbitrary additive constant, often called  constant of integration).  A primitive function is often called an  indefinite integral  (as opposed to a  definite integral  which is a mere number, not a function, usually obtained as the difference of the values of the primitive at two different points).  The usual  indefinite  notation is:

F(x)   =   ò  f (x)  dx

At a more abstract level, we may also call  Fundamental Theorem of Calculus  the generalization of the above expressed in the language of differential forms, which is also known as Stokes' Theorem, namely:

òW  dw     =     ò¶W  w

Fundamental Theorem of Calculus  (Theorem of the Day #2)   by  Robin Whitty

(2010-11-21)   Example involving complex exponentials
What is the indefinite integral of   cos(2x) e 3x  ?

That function is the real part of a complex function of a real variable:

(cos 2x  +  i sin 2x)  e 3x   =   e i (2x)  e 3x   =   e (3+2i) x

Since the derivative of  exp(a x) / a   is   exp(a x)   we obtain, conversely:

ò  e(3+2i) x dx   =   e(3+2i) x / (3+2i)   =   e3x (cos 2x  +  i sin 2x) (3-2i) / 13

The relation we were after is obtained as the  real part  of the above:

ò  cos(2x) e 3x dx   =   (3 cos 2x  +  2 sin 2x) e 3x / 13

 Brook Taylor 
 (1685-1731) (2005-02-06)   Integration by parts
One way to reduce the computation of one integral to another.

This method was first published in 1715 by  Brook Taylor (1685-1731).

The product rule states that the derivative (uv)' of a product of two functions is u'v+uv'.  When the integral of some function  f  is sought, integration by parts is a minor art form which attempts to use this backwards, by writing  f as a product u'v of two functions, one of which (u') has a known integral (u).  In which case:

ò  f dx     =     ò  u'v dx     =     uv   -  ò  uv' dx

This reduces the computation of the integral of  f  to that of  uv'.  The tricky part, of course, is to guess what choice of u would make the latter simpler...

The choice  u' = 1   (i.e., u = x  and  v = f )  is occasionally useful.  Example:

ò  ln(x) dx     =     x ln(x)  - ò (x/x) dx     =     x ln(x)  -  x

Another classical example pertains to Laplace transforms  ( p > 0 )  and/or Heaviside's operational calculus, where all integrals are understood to be  definite  integrals from 0 to +¥  (with a subexponential function  f ):

ò   f '(t) exp(-pt) dt     =     - f (0)   +    p  ò   f (t) exp(-pt) dt

Wikipedia :  Integration by parts   |   Operational calculus

rkanchan (2000-10-17)
What is the perimeter of a parabolic curve,
given the base length and height of [the] parabola?

Choose the coordinate axes so that your parabola has equation y = x2/2p for some constant parameter p. The length element  ds  along the parabola is such that (ds)2 = (dx)2 + (dy)2, or ds/dx = Ö(1+(dy/dx)2) = Ö(1 + x2/p2). The length  s  of the arc of parabola from the apex (0,0) to the point (x, y = x2/2p) is simply the following integral of this  (in which we may eliminate  x or p,  using  2py = x).

vinculum vinculum
s    =   (x/2)  Ö 1 + x2/p2   +  (p/2) ln(  Ö 1 + x2/p2  + x/p )
vinculum vinculum vinculum
  =   y Ö 1 + p/2y   +  (p/2) ln(  Ö 1 + 2y/p  +  Ö 2y/p )
vinculum vinculum
  =   (x/2) Ö 1 + (2y/x)2   +  (x2/4y) ln(  Ö 1 + (2y/x)2  + 2y/x )

For a symmetrical arc extending on both sides of the parabola's axis, the length is  2s  (twice the above).  If needed, the whole "perimeter" is  2s+2x.

 Mile-long bridge
97-Bravo (2006-06-22)   What's the top height of a (parabolic) bridge?
If a curved bridge is a foot longer than its mile-long horizontal span...

Let's express all distances in feet  (a mile is 5280 ft).  Using the notations of the previous article, 2x = 5280,  2s = 5281,  u = x/p = 2y/x = y/1320

vinculum vinculum
s / x   =   5281 / 5280  =   ½ Ö 1 + u2   +  (1/2u) ln(  Ö 1 + u2  + u )

For small values of u, the right-hand side is roughly  1+u2/6.  Solving for u the equation thus simplified, we obtain Ö(6/5280)  =  0.033709993123...  The height  y  is thus roughly equal to that quantity multiplied by  1320 ft ,  which is nearly  44.4972 ft.

This approximation is valid for any type of smooth enough curve.  It can be refined for the parabolic case using successive approximations to solve for u the above equation.  This yields  u = 0.0337128658566... which exceeds the above by about 85.2 ppm  (ppm = parts per million) for a final result of about 44.5010 ft.  The previous solution would have satisfied any engineer before the computer era.

Ronaldo  (2008-03-27; e-mail)   Length of a sagging horizontal cable:
How long is a cable which spans 28 m horizontally and sags 300 mm?

Answer :   Surprisingly, just about  28.00857 m...

Under its own weight, a uniform cable without any rigidity  (a "chain")  would actually assume the shape of a  catenary.  In a coordinate system with a vertical  y-axis and centered on its apex, the catenary has the following cartesian equation:

y/a   =   ch (x/a) -1   =   ½ (ex/a - 2 + e-x/a )   =   2 sh2 (x/2a)

Measured from the apex at  x = y = 0,  the arclength  s  along the cable is:

s   =   a  sh (x/a)

Those formulas are not easy to work with, unless the parameter  a  is given.  For example, in the case at hand  (a 28 m span with a 0.3 m sag)  we have:

x   =   14         y   =   0.3

So, we must solve for  a  (numerically) the transcendantal equation:

0.3 / a   =   2 sh2 (7/a)

This yields  a   =   326.716654425...  Therefore :

2s   =   2a sh (14 / a)   =   28.00856959...

Thus, an  8.57 mm  slack produces a  30 cm  sag for a  28 m  span.

The parameter  a  is the  radius of curvature  at the curve's apex.  If it's large enough, we may find a good approximation to the relevant transcendental equation by equating the  sh  function to its (small) argument:

y/a   =   2 sh2 (x/2a)     yields     a   »   x2 / 2y
whereby   s   =   a  sh (x/a)   »   x ( 1 + x2 / 6a2 )   »   x ( 1 + 2y2 / 3x2 )

This gives  2s   »   2x ( 1 + 8/3 (y/2x)2 )   =   28.0085714... in the above case.  This is indeed a good approximation to the aforementioned exact result.

Parabolic Approximation :

If we plug the values  x = 14  and  y = 0.3  in the above formula for the exact length of a parabolic arc, we obtain:   2s   =   28.0085690686...

Circular Approximation :

A thin circular arc of width  2x  and of height  y  has a length

2s   =     x2+y2     arcsin (   2 xy   )     =   28.00857064...
Vinculum Vinculum
y x2+y2

In fact,  all  smooth approximations to a flat enough catenary will have a comparable precision, because this is what results from equating a curve to its  osculating circle  at the lowest point.  The approximative expression we derived above in the case of the catenary is indeed quite general:

2s   »   2 x  [  1  +  8/3 (y/2x) 2  ]

abandonedmule (2001-04-10)   Arch of a Cycloid
Find the ratio, over one revolution, of the distance moved by a wheel rolling on a flat surface to the distance traced out by a point on its rim.

As a wheel of unit radius rolls on the x-axis,  the trajectory of a point on its circumference is a  cycloid,  whose parametric equation is :

x   =   t - sin(t)
y   =   1 - cos(t)

This curve was first studied by Marin Mersenne (1588-1648) around 1615.  The parameter t is the  abscissa  of the wheel's center. In the first revolution of the wheel (one arch of the cycloid),  t  goes from 0 to 2p.

The length of one full arch of a cycloid ("cycloidal arch") was first worked out in the 17th century by Evangelista Torricelli (1608-1647),  just before the advent of the calculus.  Let's do it now with modern tools:

Calling s the curvilinear abscissa (the length along the curve), we have:

(ds)2   =   (dx)2 + (dy)2   =   [(1-cos(t))2 + (sin(t))2](dt)2 
therefore:     (ds/dt)2   =   2 - 2 cos(t)   =   4 sin2(t/2)
so, if  0 ≤ t ≤ 2p:     ds/dt   =   2 sin(t/2)   ≥   0

The length of the whole arch is the integral of this when t goes from 0 to 2p and it is therefore equal to 8, [since the indefinite integral is -4 cos(t/2)]. On the other hand, the length of the trajectory of the wheel's center (a straight line) is clearly 2p (the circumference of the wheel). In other words, the trajectory of a point on the circumference is 4/p times as long as the trajectory of the center, for any whole number of revolutions (that's about 27.324% longer, if you prefer).

The ratio you asked for is the reciprocal of that, namely p/4 (which is about 0.7853981633974...), the ratio of the circumference of the wheel to the length of the cycloidal arch.  However, the result is best memorized as:

"The length of a cycloidal arch is 4 times the diameter of the wheel."

Pradip Mukhopadhyay (from Schenectady, NY.  2003-04-07; e-mail)
What is the [indefinite] integral of   (tan x)1/3 dx   ?

An obvious change of variable is to introduce   y = tan x   [ dy = (1+y2 ) dx ], so the integrand becomes  y1/3 dy / (1+y2 ).  This suggests a better change of variable, namely:  z = y2/3 = (tan x)2/3 [ dz = (2/3)y-1/3 dy ], which yields z dz = (2/3)y1/3 dy,  and makes the integrand equal to the following rational function of z, which may be integrated using standard methods (featuring a decomposition into 3 easy-to-integrate terms):

(3/2) z dz / (1+z3 )   =
¼ (2z-1) dz / (1-z+z2 )  +  (3/4) dz / (1-z+z2 )  -  ½ dz / (1+z)

As  (1-z+z2 )  is equal to the positive quantity  ¼ [(2z - 1)2 + 3] , we obtain:

ò (tan x)1/3 dx   =   ¼ ln(1-z+z2 )  +  ¼Ö3 arctg((2z-1)/Ö3)  -  ½ ln(1+z)
where  z  stands for   | tan x | 2/3

(D. B. of Grand Junction, CO. 2000-10-12)
A particle moves from right to left along the parabola y = Ö(-x) in such a way that its x coordinate decreases at the rate of 8 m/s.  When x = -4, how fast is the change in the angle of inclination of the line joining the particle to the origin?

We assume all distances are in meters.

When the particle is at a negative abscissa x, the (negative) slope of the line in question is y/x = Ö(-x)/x and the corresponding (negative) angle is thus:

a = arctg(Ö(-x)/x)
[In this, "arctg" is the "Arctangent" function, which is also spelled "atan" in US textbooks.]  Therefore, a varies with x at a (negative) rate:
da/dx = -1/(2´Ö(-x)(1-x))   (rad/m)
If x varies with time as stated, we have   dx/dt = -8 m/s,   so the angle a varies with time at a (positive) rate:
da/dt = 4/(Ö(-x)(1-x))   (rad/s)

When x is -4 m, the rate dA/dt is therefore 4/(Ö4 ´5) rad/s   =   0.4 rad/s.

The angle a, which is always negative, is thus increasing at a rate of 0.4 rad/s when the particle is 4 meters to the left of the origin (rad/s = radian per second).

 Vertically Squeezed 
 (3 to 1 ratio)
massxv2 (2002-05-18)
What's the area bounded by the following curves?
  1.   y  =  f(x)  =  x3 - 9x
  2.   y  =  g(x)  =  x  + 3

The curves intersect when f(x) = g(x), which translates into  x3 - 10x - 3 = 0. This  cubic  equation factors nicely into (x + 3) (x2 - 3x - 1) = 0 , so we're faced with only a quadratic equation...

To find if there's a "trivial" integer which is a root of a polynomial with integer coefficients [whose leading coefficient is ±1], observe that such a root would have to divide the constant term. In the above case, we only had 4 possibilities to try, namely -3, -1, +1, +3.

The abscissas   A < B < C   of the three intersections are therefore:

A  =  -3 ,     B  =  ½ (3 - Ö13)     and     C  =  ½ (3 + Ö13)

Answering an Ambiguous Question :

The best thing to do for a "figure 8", like the one at hand, is to compute the (positive) areas of each of the two lobes. The understanding is that you may add or subtract these, according to your chosen orientation of the boundary:

  • The area of the lobe from A to B (where f(x) is above g(x)) is the integral of f(x)-g(x) = x3 - 10x - 3 [whose primitive is x4/4 - 5x2 - 3x] from A to B, namely (39Ö13 - 11)/8, or about 16.202...  
  • The area of the lobe from B to C (where f(x) is below g(x)) is the integral of g(x)-f(x) from B to C, namely (39Ö13)/4, or about 35.154...

The area we're after is thus either the sum (±51.356...) or the difference (±18.952...) of these two, depending on an ambiguous boundary orientation...

If you  don't switch  curves at point B, the algebraic area may also be obtained as the integral of  g(x)-f(x)  from A to C  (up to a change of sign).

(2002-05-18, 2005-08-03)   Signed Planar Areas Consistently Defined
net  planar area is best defined as the apparent area of a 3D loop.

The area  surrounded  by a closed planar curve may be defined in general terms, even when the curve  does  cross itself many times...

The usual algebraic definition of areas depends on the orientation (clockwise or counterclockwise) given to the closed boundary of a simple planar surface. The area is positive if the boundary runs counterclockwise around the surface, and negative otherwise  (by convention, the positive direction of planar angles is always counterclockwise).  In the case of a simple closed curve  [without any multiple points]  this is often overlooked, since we normally consider only whichever orientation of the curve makes the area of its interior positive...

The clear fact that there is such an "interior" bounded by any given closed planar curve is known as "Jordan's Theorem".  It's a classical example of an "obvious" fact with a rather intricate proof.
 A curve with many 
 double points.

However, when the boundary has multiple points (like the center of a "figure 8"), there may be more than two oriented boundaries for it, since we may have a choice at a double point:  Either the boundary crosses itself or it does not (in the latter case, we make a sharp turn, unless there's an unusual configuration about the intersection).  Not all sets of such choices lead to a complete tracing of the whole loop.

 Rule   At left is the easy-to-prove "coloring rule" for a true self-crossing of the boundary, concerning the number of times the ordinary area is to be counted in the "algebraic area" dicussed here.

It's nice to consider a given oriented closed boundary as a projection of a three-dimensional loop whose apparent area is defined as a path integral.

S     =     ½   òC+   x dy - y dx     =    òC+   x dy     =     òC+   - y dx
 Positive Area 
 (Counterclockwise)  Color Codes  Positive area 
 x counterclockwise along dy  Positive area,
 -y counterclockwise along dx

brentw (Brent Watts of Hickory, NC. 2001-04-13/email)
[How do you generalize the method] of variation of parameters when solving differential equations (DE) of 3rd and higher order?
For example:   x''' - 3x'' + 4x = exp(2t)
In memory of 
Lucien Refleu

 who taught me this and
much more, many years ago.

As shown below, a high-order linear DE can be reduced to a system of first-order linear differential equations in several variables.  Such a system is of the form:

X'  =  dX/dt  =  AX + B

X is a column vector of  n  unknown functions of t.  The square matrix A  may  depend explicitely on t.  B is a vector of n explicit functions of t, called  forcing  terms.  The associated  homogeneous  system is obtained by letting  B = 0.  For a nonconstant A, it may be quite difficult to find n independent solutions of this homogeneous system  (an art form in itself)  but, once you have them, a solution of the forced system may be obtained by generalizing to n variables the method (called "variation of parameters") commonly used for a single variable.  Let's do this using only n-dimensional notations:

The fundamental object is the square matrix W formed with the n columns corresponding to the n independent solutions of the homogeneous system.  Clearly, W itself verifies the homogeneous equation:

W' = AW

It's an interesting exercise in the manipulation of determinants to prove that det(W)' = tr(A) det(W) (HINT: Differentiating just the i-th line of W gives a matrix whose determinant is the product of  det(W)  by the i-th component in the diagonal of the matrix A).  Since det(W), the so-called "Wronskian", is thus solution of a first-order linear DE, it's proportional to the exponential of some function and is therefore either nonzero everywhere or zero everywhere.  (Also, the Wronskians for different sets of homogeneous solutions must be proportional.)

Homogeneous solutions that are linearly independent at some point are therefore independent everywhere and W(t) has an inverse for any t.  We may thus look for the solution X to the nonhomogeneous system in the form  X = WY :

AX + B  =  X'  =  W'Y + WY'  =  AWY + WY'  =  AX + WY'
Therefore,   B  =  WY'

So, Y is simply obtained by integrating  W-1 B  and the general solution of the forced system may be expressed as follows, with a constant vector K  (whose n components are the n "constants of integration").  This looks very much like the corresponding formula for a single variable :

  X(t)   =   W(t)  [ K +  ò t  W-1(u) B(u) du ]  

Linear Differential Equation of Order n :

A linear differential equation of order n has the following form  (where ak and b are explicit functions of t):

x(n) + an-1 x(n-1) +  ...  + a3 x(3) + a2 x" + a1 x' + a0 x  =  b

This reduces to the above system  X' = AX + B  with the following notations :

A  =  matrix matrix matrix matrix      X  =  matrix matrix matrix matrix      B  =  matrix matrix matrix matrix
0 1 0 0  ...  0 0 x 0
0 0 1 0  ...  0 0 x' 0
0 0 0 1  ...  0 0 x" 0
0 0 0 0  ...  0 0 x(3) 0
 ...   ...   ...   ...   ...   ...   ...   ...   ... 
0 0 0 0  ...  0 1 x(n-2) 0
 -a  -a  -a  -a  ...   -an-2   -an-1  x(n-1) b
matrix matrix matrix matrix matrix matrix

The first n-1 components in the equation  X' = AX+B  merely define each component of X as the derivative of the previous one, whereas the last component expresses the original high-order differential equation.  Now, the general discussion above applies fully with a W matrix whose first line consists of n independent solutions of the homogeneous equation (each subsequent line is simply the derivative of its predecessor).

Here comes the Green function...

We need not work out every component of  W-1  since we're only interested in the first component of X...  The above boxed formula tells us that we only need the first component of W(t)W-1(u)B(u) which may be written G(t,u)b(u), by calling G(t,u) the first component of W(t)W-1(u)Z, where Z is a vector whose component are all zero, except the last one which is one.

G(t,u)  is called the  Green function  associated to the given homogeneous equation.  It has a simple expression (given below) in terms of a ratio of determinants computed for independent solutions of the homogeneous equation.  (Such an expression makes it easy to prove that the Green function is indeed associated to the equation itself and not to a particular set of independent solutions, as it is clearly invariant if you replace any solution by some linear combination in which it appears with a nonzero coefficient.)

For a third-order equation with homogeneous solutions A(t), B(t) and C(t), the expression of the Green function (which generalizes to any order) is:

G(t,u)  = 
  Determinant A(u) B(u) C(u) Determinant  
A'(u) B'(u) C'(u)
A(t) B(t) C(t)
  Determinant A(u) B(u) C(u) Determinant  
A'(u) B'(u) C'(u)
A"(u) B"(u) C"(u)

It's also a good idea to define G(t,u) to be zero when u>t, since such values of G(t,u) are not used in the integral ò t G(t,u) b(u) du. This convention allows us to drop the upper limit of the integral, so we may write a special solution of the inhomogeneous equation as the definite integral (from to , whenever it converges): ò G(t,u) b(u) du.

If this integral does not converge (the issue may only arise when u goes to ), we may still use this formal expression by considering that the forcing term b(u) is zero at any time t earlier than whatever happens to be the earliest time we wish to consider.  (This is one unsatisfying way to reestablish some kind of fixed arbitrary lower bound for the integral of interest when the only natural one, namely , is not acceptable.)

In the case of the equation x''' - 3x" + 4x = exp(2t), three independent solutions are A(t) = exp(-t), B(t) = exp(2t), and C(t) = t exp(2t). This makes the denominator in the above (the "Wronskian") equal to 9 exp(3u) whereas the numerator is (3t-3u-1)exp(u+2t)+exp(4u-t). With those values, the integral of G(t,u)exp(2u)(u)du when u goes from 0 to t turns out to be equal to f(t) = [ (9t2-6t+2)exp(2t) - 2 exp(-t) ]/54, which is therefore a special solution of your equation. The general solution may be expressed as:

x(t) = (a + bt + t2/6) exp(2t) + c exp(-t)         [ a, b and c are constant ]

Clearly, this result could have been obtained without this heavy artillery:  Once you've solved the homogeneous equation and realized that the forcing term is a solution of it, it is very natural to look for an inhomogeneous solution of the form z exp(2t) and find that z"=1/3 works.  That's far less tedious than computing and using the associated Green's function. However, efficiency in this special case is not what the question was all about...

(2001-04-13)   Convolutions and the Theory of Distributions
An introduction to the epoch-making approach of Laurent Schwartz.

  Laurent Schwartz 
The above may be dealt with using the elegant idea of convolution products among distributions.  The notorious  Theory of Distributions  occurred to the late Laurent Schwartz  (1915-2002)  "one night in 1944".  Fields Medal

For this, he received the first Fields Medal ever awarded to a Frenchman, in 1950.  Schwartz taught at  Polytechnique  from 1958 to 1980.  (He taught  me  functional analysis  in the Fall of 1977.)

A linear differential equation with constant coefficients (an important special case) may be expressed as a convolution a * x = b.  The convolution operator * is bilinear, associative and commutative.  Its identity element is the Delta distribution  d  (dubbed Dirac's "function").

Loosely speaking, the  Delta distribution  d  would correspond to a "function" whose integral is 1, but whose value at every point except zero is zero.  The integral of an ordinary function which is zero almost everywhere would necessarily be zero. Therefore, the  d  distribution  cannot possibly be an ordinary function:  Convolutions must be put in the proper context of the Theory of Distributions.  A strong case can be made that the convolution product is the notion that gives rise to the very concept of distribution.

Distributions had been used loosely by physicists for a long time, when Schwartz finally found a very simple mathematical definition for them:  Considering a (very restricted) space D of so-called  test functions, a distribution is simply a linear function which associates a scalar to every  test function.

Although other possibilities have been studied  (which give rise to less general distributions)  D is normally the so-called  Schwartz space  of infinitely derivable functions of compact support  These are perfectly smooth functions vanishing outside of a bounded domain, like the function of  x  which is  exp(-1 / (1-x 2 ))  in  [-1,+1]  and 0 elsewhere.

What could be denoted  f(g)  is written  f*g  instead.  This hint of an ultimate symmetry between the rôles of  f and g  is fulfilled by the following relation, which holds whenever the integral exists for ordinary functions f and g.

[f*g](t)   =   ò f(t-u)g(u) du

This relation may be used to establish commutativity (switch the variable to v = t-u, going from to when u goes from to ).  The associativity of the convolution product is obtained by figuring out a double integral.

Convolutions have many stunning properties.  In particular, the Fourier transform of the convolution product of two functions is the  ordinary  product of their Fourier transforms.  Another key property is that the derivative of a convolution product may be obtained by differentiating  either one  of its factors:

(f*g)'   =   f '*g   =   f*g'

This means the derivatives of a function f can be expressed as convolutions, using the derivatives of the d distribution  (strange but useful beasts):

f   =   d * f       f'   =   d' * f       f''   =  d'' * f       etc.

If the n-th order linear differential equation discussed above has constant coefficients, we may write it as  f*x = b  by introducing the distribution

f = d(n) + an-1 d(n-1) +  ...  + a3 d(3) + a2 d" + a1 d' + a0 d

Clearly, if we we have a function such that f*g=d, we will obtain a special solution of the inhomogeneous equation as g*b.  If you translate the convolution product into an integral, what you obtain is thus the general expression involving a Green function G(t,u)=g(t-u), where g(v) is zero for negative values of v.

The case where coefficients are constant is therefore much simpler than the general case: Where you had a two-variable integrator, you now have a single-variable one. Not only that, but the homogeneous solutions are well-known  (if z is an eigenvalue of multiplicity n+1 for the matrix involved, the product of exp(zt) by any polynomial of degree n, or less, is a solution).  In the important special case where all the eigenvalues are distinct, the determinants involved in the expression of G(t,u)=g(t-u) are essentially Vandermonde determinants, or Vandermonde cofactors (a Vandermonde determinant is a determinant where each column consists of the successive powers of a particular number).  The expression is thus fairly easy to work out and may be put into the following simple form, involving the characteristic polynomial P for the equation  (it's also the characteristic polynomial of the matrix we called A in the above).  For any eigenvalue z, the derivative P'(z) is the product of the all the differences between that eigenvalue and each of the others (which is what Vandermonde expressions entail):

g(v) = exp(z1v) / P'(z1) + exp(z2v) / P'(z2) + ... + exp(znv) / P'(zn)      if v>0

With this, x = g*b is indeed a special solution of our original equation f*x = b

brentw (Brent Watts of Hickory, NC. 2001-05-05)   Laplace Transforms
How do you use Laplace transforms to solve this differential system?
Initial conditions, for t=0 :   w=0, w'=1, y=0, y'=0, z= -1, z'=1.  Pierre-Simon Laplace 
  1. w" + y + z = -1
  2. w + y" - z = 0
  3. -w' -y' + z"=0

The (unilateral) Laplace transform g(p) of a function f(t) is given by:

g(p)   =   òo¥  f(t) exp(-pt) dt

This is defined, for a positive p, whenever the integral makes sense.  For example, the Laplace transform of a constant k is the function g such that g(p) = k/p.

Integrating by parts   f '(t) exp(-pt) dt   gives a simple relation, between the respective Laplace transforms L(f ') and L(f) of f ' and f :

L(f ')[p]   =   -f(0) + p L(f)[p]

This may be iterated, starting with:

L(f")[p]   =   -f '(0) + p L(f ')[p]   =   -f '(0) - p f(0) + p2 L(f)[p]

That's the basis of the so-called  Operational Calculus,  invented by Oliver Heaviside (1850-1925), which translates many practical systems of differential equations into algebraic ones.  (Originally, Heaviside was interested in the transient solutions to the simple differential equations arising in electrical circuits).

In this particular case, we may use capital letters to denote Laplace transforms of lowercase functions (W=L(w), Y=L(y), Z=L(z)...) and your differential system translates into:

  1. (p2 W - 1 - 0p)+ Y + Z = -1/p
  2. W + (p2 Y - 0 - 0p) - Z = 0
  3. -(pW - 0) -(pY - 0) + (p2 Z - 1 + p) = 0
In other words:
  1. p2 W + Y + Z = 1 -1/p
  2. W + p2 Y - Z = 0
  3. -pW -pY + p2 Z = 1-p
Solve for W,Y and Z and express the results as simple sums (that's usually the tedious part, but this example is designed to be simpler than usual):
  • W = 1/(p2 +1)
  • Y = p/(p2 +1) - 1/p
  • Z = 1/(p2 +1) - p/(p2 +1)2
The last step is to go from these Laplace transforms back to the original (lowercase) functions of t, with a reverse lookup using a table of Laplace transforms, similar to the (short) one provided below.
  • w = sin(t)
  • y = cos(t) - 1
  • z = sin(t) - cos(t)

With other initial conditions, solutions may involve various linear combinations of no fewer than 5 different types of functions (namely: sin(t), cos(t), exp(-t), t and the constant 1), which would make a better showcase for Operational Calculus than this particularly simple example...

Below is a small table of Laplace transforms, which is sufficient to solve the above for any set of initial conditions, by  reverse lookup :

Laplace transform
g(p) = òo¥ f(t) exp(-pt) dt
1  =  t 01/p
t nn! / pn+1  =  G(n+1) / pn+1
t ½  =  Öt½ Öp / p3/2
exp(at)1 / (p-a)
sin(kt)k / (p2 + k2 )
cos(kt)p / (p2 + k2 )
exp(at) sin(kt)k / ([p-a]2 + k2 )
exp(at) cos(kt)[p-a] / ([p-a]2 + k2 )
sin(t) / tp/2 - Arctg p
d [Dirac Delta]1
f '(t)p g(p)  -  f(0)
f ''(t)p2 g(p)  -  f(0) -  p f '(0)

brentw (Brent Watts of Hickory, NC. 2001-03-19)   Counterexamples
1)   What's an example of a function  f  for which the integral
from to is well-defined for  |f(x)| dx  but not for  f(x) dx.
2)   What's an example of a function  f  for which the opposite is true?

1) Consider any  nonmeasurable  set E within the interval [0,1]  (Zermelo's Axiom of Choice guarantees one such set exists)  and define f(x) to be:

  • +1 if x is in E
  • -1 if x is in [0,1] but not in E
  • 0 if x is outside [0,1]

The function f is not Lebesgue-integrable, but its absolute value clearly is (|f(x)| is equal to 1 on [0,1] and 0 elsewhere).

That was for Lebesgue integration. For Riemann integration, you may construct a simpler example by letting the above E be the set of rationals between 0 and 1.

2) On the other hand, the function sin(x)/x is a simple example of a function which is Riemann-integrable over ]-¥,+¥[ (Riemann integration can be defined over an infinite interval, although it's not usually done in basic textbooks), whereas the absolute value |sin(x)/x| is not. Neither function is Lebesgue-integrable over ]-¥,+¥[, although both are over any finite interval.

araimi (2001-04-10)   Functional Calculus
Show that: f (D)[eax y] = eax f (D+a)[y] , where D is the operator d/dx.

The notation pertains to  functional analysis.  Let's explain:

If  f (x) is the converging sum of all terms  axn  (for some scalar sequence a), f  is called an analytic function [about zero] and it can be defined for some nonnumerical things that can be added, scaled or "exponentiated"...

The possibility of exponentiation to the power of a nonnegative integer reasonably requires the definition of some kind of multiplication with a neutral element  (in order to define the zeroth power)  but that multiplication need not be commutative or even associative.  Power-associativity suffices  (which is implied, in particular, by the  alternativity  obeyed by  octonions).  Here we merely focus on the multiplication of square matrices of finite sizes which corresponds to the composition of linear functions in a vector space of finitely many dimensions.

If  M  is a finite square matrix representing some linear operator (which we shall denote by the same symbol  M  for convenience)  f (M) is defined as a power series of M.  If there's a vector basis in which the operator  M  is diagonal,  f (M) is diagonal in that same basis, with f (z) appearing on the diagonal of f (M) wherever z appears in the diagonal of M.

Now, the differential operator D is a linear operator like any other, whether it operates on a space of finitely many dimensions (for example, polynomials of degree 57 or less) or infinitely many dimensions (polynomials, formal series...).  f (D) may thus be defined the same way. It's a formal definition which may or may not have a numerical counterpart, as the formal series involved may or may not converge. The same thing applies to any other differential operator, and this is how  f (D) and f (D+a)  are to be interpreted.

To prove that a linear relation holds when  f  appears homogeneously (as is the case here), it is enough to prove it for any n when  f (x)=xn :

  • The relation is trivial for n=0 (the zeroth power of any operator is the identity operator) as the relation translates into exp(ax)y = exp(ax)y.
  • The case n=1 is: D[exp(ax)y] = a exp(ax)y + exp(ax)D[y] = exp(ax)(D+a)[y].
  • The case n=2 is obtained by differentiating the case n=1 exactly like the case n+1 is obtained by differentiating case n, namely:
    Dn+1[exp(ax)y] = D[exp(ax)(D+a)n(y)]
    = a exp(ax)(D+a)n[y] + exp(ax) D[(D+a)n(y)]
    = exp(ax) (D+a)[(D+a)n(y)] = exp(ax) (D+a)n+1[y].

This completes a proof by induction for any f (x) = xn, which establishes the relation for any analytic function f, through summation of such elementary results  (when such a summation  makes sense).

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