By M. Bocher
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Additional info for An Introduction to the Study of Integral Equations
On differentiation with respect to x this becomes 2x + 2 y dy = 0; dx y x FIGURE 2 13 The Nature of Differential Equations and since c is already absent, there is no need to eliminate it and x+y dy =0 dx (5) is the differential equation of the given family of circles. Similarly, x2 + y2 = 2cx (6) is the equation of the family of all circles tangent to the y-axis at the origin (Figure 3). When we differentiate this with respect to x, we obtain dy = 2c dx 2x + 2 y or x+y dy =c dx (7) y x FIGURE 3 14 Differential Equations with Applications and Historical Notes The parameter c is still present, so it is necessary to eliminate it by combining (6) and (7).
The situation discussed here is an example of exponential decay. This phrase refers only to the form of the function (8) and the manner in which the quantity x diminishes, and not necessarily to the idea that something or other is disintegrating. x x0 ½ x0 T FIGURE 6 t 24 Differential Equations with Applications and Historical Notes Example 4. Mixing. A tank contains 50 gallons of brine in which 75 pounds of salt are dissolved. Beginning at time t = 0, brine containing 3 pounds of salt per gallon flows in at the rate of 2 gallons per minute, and the mixture (which is kept uniform by stirring) flows out at the same rate.
Dt If we write this differential equation for A in the form dA A = k, dt then we see that k can be thought of as the fractional change in A per unit time, and 100k is the percentage change in A per unit time. Example 2. Population growth. Suppose that x0 bacteria are placed in a nutrient solution at time t = 0, and that x = x(t) is the population of the colony at a later time t. 8 Since the rate of increase of x is proportional to x itself, we can write down the differential equation dx = kx . dt By separating the variables and integrating, we get dx = k dt , x log x = kt + c.