As t→∞, the population asymptotically approaches Pmax. Note that the population grows quickly at first, but the rate of increase slows as the population reaches the maximum. Figure 7 shows the graph of a typical solution. t P P = Pmax 1 + Ce-kt Pmax Figure 7: Logistic population growth. Solving for P gives P = Pmax 1 + Ce−kt where the value of C changed. We can use a logarithm rule to combine the two terms on the left: ln ∣∣∣∣ PPmax − P ∣∣∣∣ = kt + C so P Pmax − P = Cekt. The result is ln |P | − ln |Pmax − P | = kt + C. Each of the simpler fractions can then be integrated easily. This is a simple example of the integration technique known as partial fractions decomposition. The secret is to express the fraction as the sum of two simpler fractions: Pmax P (Pmax − P ) = 1 P + 1 Pmax − P. The integral on the left is difficult to evaluate. We can now separate to get∫ Pmax P (Pmax − P ) dP = ∫ k dt. We begin by multiplying through by Pmax Pmax dP dt = kP (Pmax − P ). This circuit has two components: APPLICATIONS OF DIFFERENTIAL EQUATIONS 5 We can solve this differential equation using separation of variables, though it is a bit difficult. EXAMPLE 3 Figure 4 shows a simple kind of electric circuit known as an RC circuit. The following example illustrates a more complicated situation where the natural growth equation arises. The doubling time can be obtained by substituting y = 2y0 and then solving for t. the amount of time that it takes for y to grow to twice its original value. Similarly, given a growing variable y = y0e kt (k > 0) we can measure the rate of exponential growth using the doubling time, i.e. The half life can be obtained by substituting y = y0/2 y0 2 = y0e −rt and then solving for t. If k 0 t y y = e kt k 0) the half life is the amount of time that it takes for y to decrease to half of its original value.If k > 0, then the variable y increases exponentially over time.The sign of k governs the behavior of the solutions: The constant k is called the rate constant or growth constant, and has units of inverse time (number per second). Its solutions have the form y = y0e kt where y0 = y(0) is the initial value of y. THE NATURAL GROWTH EQUATION The natural growth equation is the differential equation dy dt = ky where k is a constant. Exponential Growth and Decay Perhaps the most common differential equation in the sciences is the following. M for mass, P for population, T for temperature, and so forth. In addition, the letter y is usually replaced by a letter that represents the variable under consideration, e.g. For such a system, the independent variable is t (for time) instead of x, meaning that equations are written like dy dt = t3y2 instead of y′ = x3y2. systems that change in time according to some fixed rule. The most common use of differential equations in science is to model dynamical systems, i.e. chemical reactions, population dynamics, organism growth, and the spread of diseases. Almost all of the known laws of physics and chemistry are actually differential equa- tions, and differential equation models are used extensively in biology to study bio-A mathematical model is a description of a real-world system using mathematical language and ideas. Download Applications of Differential Equations and more Differential Equations Study Guides, Projects, Research in PDF only on Docsity!3 Applications of Differential Equations Differential equations are absolutely fundamental to modern science and engineering.
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