The differential equation is dP dt rP 1 P M P is the population r is the

# The differential equation is dp dt rp 1 p m p is the

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The differential equation is dP dt = rP 1 - P M P is the population, r is the Malthusian rate of growth, and M is the carrying capacity of the population This is a first order , nonlinear , homogeneous differential equation We solve this problem later in the semester Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (28/47)

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The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Applications of Differential Equations 5 The van der Pol Oscillator: In electrical circuits, diodes show a rapid rise in current, leveling of the current, then a steep decline Biological applications include a similar approximation for nerve impulses The van der Pol Oscillator satisfies the differential equation v 00 + a ( v 2 - 1) v 0 + v = b v is the voltage of the system, and a and b are constants This is a second order , nonlinear , nonhomogeneous differential equation This problem does not have an easily expressible solution, but shows interesting oscillations Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (29/47)
The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Applications of Differential Equations 6 Lotka-Volterra – Predator and Prey Model: Model for studying the dynamics of predator and prey interacting populations Model for the population dynamics when one predator species and one prey species are tightly interconnected in an ecosystem System of differential equations x 0 = a x - b xy y 0 = - c y + d xy x is the prey species, and y is the predator species This is a system of first order , nonlinear , homogeneous differential equations No explicit solution, but we’ll study its behavior Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (30/47)

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The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Applications of Differential Equations 7 Forced Spring-Mass Problem with Damping: An extension of the spring-mass problem that includes viscous-damping caused by resistance to the motion and an external forcing function that is applied to the mass The model is given by my 00 + cy 0 + ky = F ( t ) y is the position of the mass, m is the mass of the object, c is the damping coefficient, k is the spring constant, F ( t ) is an externally applied force This is a second order , linear , nonhomogeneous differential equation We’ll learn techniques for solving this Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (31/47)
The Class — Overview The Class...

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