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Chapter 1 First-order Differen-tial Equations 1.1 Introduction Example 1.1.1 Dating of art works On May 29�� 1945�� H.A.Van Meegeren�� a third rate Dutch painter[3]�� was arrested on the charge of collaborating with the enemy for [4] the sale to Goering of a painting of famed 17th century Dutch painter. Van Meegeren refused to accept the charge and an- nounced�� in his prison cell [5]�� that he had never sold painting to Goering. He stated that all the questioned paintings[6] were his own works. To settle the question an international panel of dis- tinguished chemists�� physicists and art historians was appointed to investigate the matter. The panel took X-rays of the paint- ings to determine whether other paintings were underneath those paintings�� analyzed the pigments[7] (coloring materials) used in the paintings�� and examined the paintings for certain signs of old age[8]. The panel of experts found traces of the modern pigment cobalt blue[9] in some paintings. In addition�� they also detected phenoformaldehyde[10]�� which was not discovered until the turn of the 19th century[11]�� in several paintings. On the basis of these evidences Van Meegeren was convicted�� of forgery[12]�� on October 12�� 1947 and sentenced to one year in prison[13]. Two months later�� he died of a heart attack. However�� many people refused to believe that the famed \Dis- ciples at Emmaus[14]" was a forgery[15]. In 1967�� almost twenty years later�� scientists at Carnegie Mellon University proved that the \Disciples at Emmaus" was indeed a forgery. The key to the dating of materials lies in the phenomenon of radioactivity[16] discovered at the turn of the 20th century by the physicist Rutherford and his colleagues. They showed that the atoms of certain \radioactive" elements are unstable[17] and that within a given time period a fixed proportion of the atoms spontaneously disintegrates to form atoms of a new element[18]. Rutherford also showed that the radioactivity of a substance is directly proportional to[19] the number of atoms of the substance present. Let N(t) denotes the number of atoms present at time t�� then dN/dt�� the number of atoms that disintegrate per unit time�� is proportional to N�� thus we have the following equation (1.1.1) where[20] constant �� is positive and is known as the decay con- stant of the substance[21]. Usually�� we use half-life[22]�� the time required for half of a given quantity of radioactive atoms to decay�� to measure the rate of disintegration of a substance. Assume that N(t0) = N0�� then we have the mathematical model for computing half-life (1.1.2) By evaluating the present disintegration rates of the radioac- tive pigments in Van Meegeren's questioned paintings�� the experts concluded that the paintings \Disciples at Emmaus"�� \Woman Reading Music[23]" and \Woman Playing Mandolin[24]" must be modern forgeries. Example 1.1.2 Detection of diabetes[25] Diabetes mellitus[26] is a disease of metabolism which is charac- terized by too much sugar in the blood and urine[27]. Glucose tolerance test[28] (GTT) is a commonly used method to diagnose the disease. In this test�� the patient is asked to take a large dose of glucose after an overnight fast[29]. During the next three to five hours�� several measurements of the concentration of glu- cose are made in the patient's blood�� and these measurements are used in the diagnosis of diabetes. Unfortunately�� there is no universally accepted criterion[30] exist for interpreting the results of a GTT. Different physicians[31] interpreting the results of a GTT may come up with[32] different diagnoses. Here is a case. A Rhode Island physician�� after reviewing the results of a GTT�� came up with a diagnosis of diabetes. But another physician de- clared the patient to be normal after reviewing the results of the same GTT. To settle the question�� the results of the GTT were sent to a specialist in Boston. After examining these results�� the specialist concluded that the patient was suffering from a pituitary tumor[33]. In the mid-1960's�� Drs. Rosevear and Molnar of the Mayo Clinic and Drs. Ackerman and Gatewood of the University of Min- nesota discovered a fairly reliable criterion for interpreting the re- sults of a GTT. They constructed a model which could accurately describe the blood glucose regulatory system[34] during a glucose tolerance test and in which one or two parameters[35] would yield criteria for distinguishing normal individuals from mild diabetics and prediabetics[36]. The basic model is described analytically[37] by following sys- tem of equations[38] (1.1.3) (1.1.4) where G denotes the concentration of glucose in the blood and H denotes the concentration of the net hormonal[39] concentration. The function J(t) is the external rate at which the blood glucose concentration is being increased. Definition 1.1.1 An equation relating an unknown function�� its derivatives[40] and independent