9 Differential Equations

9.1 Modeling with Differential Equations

Models of Population Growth

One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population. the variables are: t = time (the independent variable) P = the number of individuals in the population (the dependent variable) Then the rate of growth is: where k is the proportionality constant

Solution of this differential equation:

Models of Population Growth This model is appropriate for population growth under ideal conditions, but we have to recognize that a more realistic model must reflect the fact that a given environment has limited resources. Many populations start by increasing in an exponential manner, but the population levels off when it approaches its carrying capacity M (or decreases toward M if it ever exceeds M).

Models of Population Growth For a model to take into account both trends, we make two assumptions:  Initially: if P is small (the growth rate is proportional to P.)  Later: if P > M (P decreases if it ever exceeds M so growth rate becomes negative.) A simple expression that incorporates both assumptions is given by the equation: “ “ logistic differential equation”

Models of Population Growth If P is small compared with M, then P/M is close to 0 and so dP/dt  kP. If P > M, then (1 – P/M) is negative and so dP/dt < 0. Notice that the constant functions P(t) = 0 and P(t) = M are solutions These two constant solutions are called equilibrium solutions.

Models of Population Growth If the initial population P(0) lies between 0 and M, then the right side of is positive, so dP/dt > 0 and the population increases. But if the population exceeds the carrying capacity (P > M), then 1 – P/M is negative, so dP/dt < 0 and the population decreases. So in either case, if the population approaches the carrying capacity (P  M), then dP/dt  0, which means the population levels off.

Solution of the logistic equation: Solutions of the logistic equation

A Model for the Motion of a Spring

Let’s now look at an example of a model from the physical sciences. We consider the motion of an object with mass m at the end of a vertical spring

A Model for the Motion of a Spring Hooke’s Law says that if the spring is stretched (or compressed) x units from its natural length, then it exerts a force that is proportional to x: restoring force = –kx where k is a positive constant (called the spring constant). If we ignore any external resisting forces (due to air resistance or friction) then, by Newton’s Second Law (force equals mass times acceleration), we have

A Model for the Motion of a Spring This is an example of what is called a second-order differential equation because it involves second derivatives. Let’s see what we can guess about the form of the solution directly from the equation. We can rewrite it in the form: which says that the second derivative of x is proportional to x but has the opposite sign. Solve by integrating twice.

General Differential Equations

Definition A differential equation is an equation containing an unknown function and its derivatives. Examples: y is the dependent variable and x is independent variable Ordinary Differential Equations

Partial Differential Equation Examples: u is the dependent variable and x and y are independent variables. u is dependent variable and x and t are independent variables

Order of a Differential Equation The order of the differential equation is the order of the highest derivative in the differential equation. Differential Equation ORDER 1 2 3

Degree of Differential Equation Differential Equation Degree The degree of a differential equation is the power of the highest order derivative term in the differential equation.

Linear Differential Equation (section 9.5) non-linear Examples : linear A linear differential equation is any differential equation that can be written in the following form: The important thing to note about linear differential equations is that there are no products of the function, and its derivatives and neither the function or its derivatives occur to any power other than the first power. non-linear linear non-linear

21 Solving Differential Equations

The most general first order differential equation can be written as: There is no general formula for the solution. We will look at two types of these and how to solve them: Separable EquationsSeparable Equations (9.3) Linear EquationsLinear Equations (9.5) First Order Differential Equations

Differential Equations: Separable Equations A separable differential equation is any differential equation that we can write in the following form Now rewrite the differential equation as: Integrate both sides. Use the initial condition to find the constant of integration.

Solve with y(0) = 1. Practice Example: Step1: Divide through by y. We get: Step2: Integrate both sides: Step3: Solving for y gives: where Step4: Use the initial condition: to get: A = 1 So the solution to the problem is: