Section 1.3: Differential Equations As Mathematical Models a Mathematical Model Is a Mathematical Description of a Physical Syst

Section 1.3: Diﬀerential Equations as Mathematical Models

A mathematical model is a mathematical description of a physical system or phenomenon.

Dynamical System:

A system that changes or evolves in time t. • Consists of a set of time-dependent variables, called state variables • and rules (i.e. mathematical model) for determining the state of the

system at any time in terms of a speciﬁed state at some time t0.

The state of the system at time t is the value of the state variables • at t; the speciﬁed state at time t0 is the initial condition.

Classiﬁed as either discrete-time systems or continuous-time systems, • dependent on whether the state variables are deﬁned at discrete times or over a continuous time interval.

For a continuous-time dynamical system, the mathematical model (or • rule) is a DE or system of DEs, and the speciﬁed state of the system at

time t0 is the initial condition. This gives an IVP.

The solution of the IVP is called the response of the system. •

Static systems are systems that do not evolve in time.

1 Steps in Model Development:

1. Make a clear formulation of the real system or phenomenon to be mod- eled, identifying the variables that control the system.

2. Choose the variables to include in the model (i.e., specify the level of resolution).

3. Make reasonable assumptions about the system, including any applicable empirical laws. There are diﬀerent levels of assumptions.

4. Formulate the model. This often consists of a set of ODEs and/or PDEs, as well as side conditions (e.g., initial conditions, boundary conditions).

5. Solve the set of derived equations. This can be diﬃcult and most often requires numerical techniques.

6. Compare model predictions with reality. Make comparisons in terms of known behavior or trends, or common sense (qualititative comparison). Whenever possible, make comparisons with measured data from experiments (quantitative comparison).

7. If good agreement, then use the model (until it is proven bad); If unsatisfactory agreement, then modify level of resolution or assumptions. That is, go back to step 2 or 3.

2 Example: Falling Bodies and Air Resistance

A body of weight w falls from rest under the inﬂuence of gravity and a retarding force resulting from air resistance. It is assumed that the force du to air resistance is proportional to the velocity of the body.

Find an ODE for the velocity of the body at any time t. Then formulate an appropriate IVP.

Recall: Newton’s Second Law of Motion:

F = m a where F = sum of forces acting on a body m = mass of the body a = acceleration of the body

Let s = distance the body has fallen from its initial position

3 Example: Falling Bodies and Air Resistance (cont)

From Calculus: ds Velocity: v(t) = dt dv d2s Acceleration: a(t) = = dt dt2

Forces acting on body:

1. Gravity: acts downward, that is, in the positive direction, and is equal to the weight of the body

F1 = w = m g

where g is the acceleration due to gravity

2. Air resistance: acts upward, that is, in the negative direction, and is assumed to be proportional to the velocity of the body

F = k v 2 − where k > 0 is a constant

4 Example: Falling Bodies and Air Resistance (cont)

Total force on the body:

F = F + F = m g k v 1 2 −

Therefore, Newton’s Second Law of Motion (F = m a) gives:

dv m g k v = m − dt

dv k = = g v ⇒ dt − m

This is a ﬁrst-order, linear ODE.

IVP: dv k = g v, t > 0 dt − m

v(0) = 0

5 Example: Mixtures (Problem 10, page 28)

Suppose a large mixing tank initially holds 300 gallons of water into which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min and, when the solution is well stirred, it is pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering the tank is 2 lbs/gal, determine a diﬀerential equation for the amount A(t) of salt in the tank at time t.