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What is a Differential Equation? ... PDF Print E-mail
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Tuesday, 12 January 2010 17:30

Second order nonlinear ordinary differential equation describing the motion of a pendulum of length L

According to common University books, "A Differential Equation is equation of more or dependent variables with respect to one or more independent variables" (Shepley L. Ross - Introduction to Differential Equations).

I prefer to be more explicit, "A Differential Equation is an equation linear or non linear where the variables are the differential of any degree, with it relative coefficients".

Differential Equations are valid for any purpose to describe motion, including motion also the speed of grow of a battery or the decay of chemical components.

In fact, the book, Martin Braun - Differential Equations and Their Applications 3d offers several interesting examples.

I will include here one example, that he consider speaking about  The Van Meegeren art forgeries at page 24.

1.3 The Van Meegeren art forgeries

After the liberation of Belgium in World War II, the Dutch Field Security began its hunt for Nazi collaborators. They discovered, in the records of a firm which had sold numerous works of art to the Germans, the name of a banker who had acted as an intermediary in the sale to Goering of the painting "Woman Taken in Adultery" by the famed 17th century Dutch painter Jan Vermeer.

The banker in turn revealed that he was acting on behalf of a third rate Dutch painter H. A. Van Meegeren, and on May 29, 1945 Van Meegeren was arrested on the charge of collaborating with the enemy. On July 12, 1945 Van Meegeren startled the world by announcing from his prison cell that he had never sold "Woman Taken in Adultery" to Goering. Moreover, he stated that this painting and the very famous and beautiful "Disciples at Emmaus", as well as four other presumed Vermeers and two de Hooghs (a 17th century Dutch painter) were his own work. Many people, however, thought that Van Meegeren was only lying to save himself from the charge of treason. To prove his point, Van Meegeren be­gan, while in prison, to forge the Vermeer painting "Jesus Amongst the Doctors" to demonstrate to the skeptics just how good a forger of Vermeer he was. The work was nearly completed when Van Meegeren learned that a charge of forgery had been substituted for that of collaboration. He, therefore, refused to finish and age the painting so that hopefully investiga­tors would not uncover his secret of aging his forgeries. To settle the ques­tion an international panel of distinguished chemists, physicists and art historians was appointed to investigate the matter. The panel took x-rays of the paintings to determine whether other paintings were underneath them. In addition, they analyzed the pigments (coloring materials) used in the paint, and examined the paintings for certain signs of old age.

Now, Van Meegeren was well aware of these methods. To avoid detec­tion, he scraped the paint from old paintings that were not worth much, just to get the canvas, and he tried to use pigments that Vermeer would have used. Van Meegeren also knew that old paint was extremely hard, and impossible to dissolve. Therefore, he very cleverly mixed a chemical, phenoformaldehyde, into the paint, and this hardened into bakelite when the finished painting was heated in an oven.

However, Van Meegeren was careless with several of his forgeries, and the pane! of experts found traces of the modern pigment cobalt blue. In addition, they also detected the phenoformaldehyde, which was not dis­covered until the turn of the 19th century, in several of the paintings. On the basis of this evidence Van Meegeren was convicted, of forgery, on Oc­tober 12, 1947 and sentenced to one year in prison. While in prison he suffered a heart attack and died on December 30, 1947.

However, even following the evidence gathered by the panel of experts, many people still refused to believe that the famed "Disciples at Emmaus" was forged by Van Meegeren. Their contention was based on the fact that the other alleged forgeries and Van Meegeren's nearly completed "Jesus Amongst the Doctors" were of a very inferior quality. Surely, they said, the creator of the beautiful "Disciples at Emmaus" could not produce such in­ferior pictures. Indeed, the "Disciples at Emmaus" was certified as an authentic Vermeer by the noted art historian A. Bredius and was bought by the Rembrandt Society for $170,000. The answer of the panel to these skeptics was that because Van Meegeren was keenly disappointed by his lack of status in the art world, he worked on the "Disciples at Emmaus" with the fierce determination of proving that he was better than a third rate painter. After producing such a masterpiece his determination was gone. Moreover, after seeing how easy it was to dispose of the "Disciples at Emmaus" he devoted less effort to his subsequent forgeries. This explana­tion failed to satisfy the skeptics. They demanded a thoroughly scientific and conclusive proof that the "Disciples at Emmaus" was indeed a forgery. This was done recently in 1967 by scientists at Carnegie Mellon University, and we would now like to describe their work.

The key to the dating of paintings and other materials such as rocks and fossils lies in the phenomenon of radioactivity discovered at the turn of the century. The physicist Rutherford and his colleagues showed that the atoms of certain "radioactive" elements are unstable and that within a given time period a fixed proportion of the atoms spontaneously disin­tegrates to form atoms of a new element. Because radioactivity is a prop­erty of the atom, Rutherford showed that the radioactivity of a substance is directly proportional to the number of atoms of the substance present. Thus, if 7V(f) denotes the number of atoms present at time /, then dN/di, the number of atoms that disintegrate per unit time is proportional to N, that is,

dN/dt = - λN

The constant  λ which is positive, is known as the decay constant of the substance. The larger λ is, of course, the faster the substance decays. One measure of the rate of disintegration of a substance is its half-life which is defined as the time required for half of a given quantity of radioactivity atoms to decay.

After some calculation (very simple) that you can read on the book:

 

We can deduce that the half-life of a substance is ln2 divided by the decay constant λ.

Now, all paintings contain a small amount of the radioactive element lead-210 (2l0Pb), and an even smaller amount of radium-226 (226Ra), since these elements are contained in white lead (lead oxide), which is a pigment that artists have used for over 2000 years. For the analysis which follows, it is important to note that white lead is made from lead metal, which, in turn, is extracted from a rock called lead ore, in a process called smelting. In this process, the lead-210 in the ore goes along with the lead metal. However, 90-95% of the radium and its descendants are removed with other waste products in a material called slag.
Thus, most of the supply of lead-210 is cut off and it begins to decay very rapidly, with a half-life of 22 years. This process continues until the lead-210 in the white lead is once more in radioactive equilibrium with the small amount of radium present, i.e. the disintegration of the lead-210 is exactly balanced by the disintegration of the radium.

... You can read the complete story in the book.

And therefore, Differential Equations can be used to discover modern forgery, because the material used in the paints remains to "new".

Professor Braun also offers a method in 2.7 page 172, "A model for the detection of diabetes".

Thanks,

Giovanni A. Orlando.

PS. I apologize for the shortage of this introduction to differential equations, that I expect to be sufficient. The reason for my short description is because I am close to complete the description of the book "Telos, Vol 1", and to include the chapters about Immortality and the awakening of the 32 DNA Layers.

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Last Updated on Tuesday, 12 January 2010 17:49