Examples of First-Order Differential Equations
Phenomena in many disciplines are modeled by first-order differential equations. Some examples include
Mechanical Systems
Consider a ball of mass m falling under the influence of gravity. Let y(t) denote the height of the ball and v(t) denote the velocity of the ball. (In our coordinate system positive is upward.) According to Newton's law, mass times acceleration equals force, we get the following differential equations:
The first equation can be simplified to read v'=-g. These two differential equations can be accompanied by initial conditions: the initial position y(0) and velocity v(0).
The above problem can be solved easily. The equations are both directly integrable. If you have taken physics class you should remember that
where v_0 is the initial velocity and y_0 is the initial position. Suppose we now assume that the ball is subject to air resistance. A simple model is that the force of air resistance is proportional to velocity. In this case, the new differential equations are
The term -kv(t) represents air resistance and k is a constant. The minus sign means that air resistance acts in the direction opposite to the motion of the ball. It is more difficult to solve this problem exactly. So we will not give the solution here. This is an example of a linear ode.
Electrical Circuits
Consider a series RC (resistor and capacitor in series) circuit with voltage source V(t). Let I(t) denote the current. The differential equation for the current is
Here R is the resistance of the resistor and C is the capacitance of the capacitor (both are constants). This is another example of a linear ode.
Population Models
The simplest population growth model, the Malthusian model, states that the rate of change of population is proportional to the population. In mathspeak we have
Here P(t) is the population at time t and a is a constant.
(We assume that the population is a continuous function for
simplicity. If we assume P(t) represents number of people,
then obviously P(t) can take only integer values. A different
type of analysis is required.)
This is an example of
a linear separable
ode.
The exact solution to the problem is
If a is positive, the populations grows exponentially for all time. This is unrealistic. A more realistic model is the logistic model
Here a and b are constants. In this model P'(t) is a sum of positive and negative terms (assuming P(t) is non-negative). If P(t) is sufficiently large, P'(t) is negative. It turns out as t increases the poplulation approaches b. b is the carrying capacity of this model. This is an example of a nonlinear separable ode.
Newton's Law of Cooling
Let T(t) denote the temperature of an object and let M(t) be the temperature of the surrounding environment. Newton's law of cooling states that the rate of change of temperature of the object is proportional to the difference between the object and environment temperatures. The differential equation is
Here k is a positive constant. Notice the negative sign. If the object is at a higher temperature than the environment, then T'(t) is negative and the temperature decreases, agreeing with intuition. This is another example of a linear ode.
Compartmental Analysis
Consider a tank with volume 100L containing a salt solution. Suppose a solution with 2kg/L of salt flows into the tank at a rate of 5L/min. The solution in the tank is well-mixed. Solution flows out of the tank at a rate of 5L/min. If initially there is 20kg of salt in the tank, how much salt will be in the tank as a function of time?
This is an example of compartment problem. Let x(t) be the amount of salt in the tank in kg. x'(t) is equal to the rate at which salt enters the tank minus the rate at which salt leaves the tank. The rate at which salt enters the tank is (5L/min)(2kg/L)=10kg/min. The rate at which salt leaves the tank is equal the rate flow of solution out of the tank times the concentration of salt in the solution. The concentration of salt is x(t)/100L. Hence the rate of outflow of salt is (5L/min)(x(t)/100L). The differential equation for the amount of salt is
The is another example of a linear ode. The solution of the ode is
