- Identify linear first order differential equations.
- Model behavior of certain systems using first order linear differential equations.
- Use the input signal and system response paradigm to obtain an ODE for a physical system.
- Check reasonableness of models using unit analysis .
Definition 3.2 An initial value problem is a differential equation together with initial conditions.
4. Cell division
Here we will see how the differential equation for our secret function appears when modeling a natural phenomenon – the population growth of a colony of cells…In this example we’ll model the number of yeast cells in a batch of dough. As we work through this example, pay careful attention to the assumptions we make, and how the initial condition plays a role in the resulting differential equation.
For our system, we assume we have a colony of yeast cells in a batch of bread dough. The first step is to identify the variables, the units, and give them names.
|number of cells|
|time measured in seconds|
We also need to set some initial condition, y0, the number of cells that we begin with at t=0. In this system, this might be the number of yeast cells in a yeast packet.
A differential model
If y denotes the number of yeast cells, what can we say about the derivative y˙? The derivative represents the rate at which the number of cells is growing. How should it depend on the number of cells? In nature, cells given plenty of space and food tend to divide through mitosis regularly. If we assume that each cell is dividing independently of all other cells, then doubling the number of cells should double the rate at which new cells are born. In fact, multiplying the number of cells by any scalar factor should do the same to the derivative. So this directly implies that the growth rate of cells is proportional to the number of cells:
We can make this into a true equation by simply inserting a proportionality constant a, such that
We say that 1/a is a “characteristic” timescale for our problem, setting the rate at which the cells divide. A solution to the above differential equation is
where y0 is the number of yeast cells we started with at t=0. In our case, we assume that y0 is the number of yeast cells in a packet, which is about 180 billion yeast cells.
5. Classification of differential equations
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There are two kinds:
- An ordinary differential equation (ODE) involves derivatives of a function of only one variable.
- A partial differential equation (PDE) involves partial derivatives of a multivariable function.
When we consider ODEs, we will often regard the independent variable to be time…The dot notation y˙ should only be used to refer to a time derivative. If for example y is a function of a spacial variable y=y(x), we will only use the notation y′ to denote the derivative with respect to x.
Definition 5.1 The order of a DE is the highest n such that the nth derivative of the function appears…
The order is 5, because the highest derivative that appears is the 5th derivative, y(5).
7. Natural growth and decay equations..We’ve been introduced to a few basic forms of differential equations so far. The first equation we saw was a basic growth equation,
which, when a is a positive constant, governs systems like bank accounts and cell populations. If we put a negative sign in front of a we get the decay equation
which can be used to describe things like radioactive decay of materials.
How would you classify the differential equations y˙=ay and y˙=−ay just discussed? Choose all descriptors that apply…
These two equations are both first order, linear, homogeneous differential equations. To see that these equations are homogeneous, we can either check that y=0 is a solution (it is), or we can rewrite them in standard linear form:
8. Introduction to modeling
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There are two kinds of modeling. We’re not going to talk about the kind that involves fancy clothes and photographs. The other kind, mathematical modeling , is converting a real-world problem into mathematical equations.
- Identify relevant quantities, both known and unknown, and give them symbols. Find the units for each.
- Identify the independent variable(s). The other quantities will be functions of them, or constants. Often time is the only independent variable.
- Write down equations expressing how the functions change in response to small changes in the independent variable(s). Also write down any “laws of nature” relating the variables. As a check, make sure that all summands in an equation have the same units.
Often simplifying assumptions need to be made; the challenge is to simplify the equations so that they can be solved but so that they still describe the real-world system well.
I have a savings account earning interest compounded daily, and I make frequent deposits or withdrawals into the account. Find an ODE with initial condition to model the balance.
- Daily compounding is almost the same as continuous compounding, so let’s assume that interest is paid continuously instead of at the end of each day.
- Similarly, let’s assume that my deposits/withdrawals are frequent enough that they can be approximated by a continuous money flow at a certain rate, the net deposit rate (which is negative when I am withdrawing).
Variables and functions (with units): Define the following:
the initial amount that the account starts with (dollars)
time from the start (years)
the interest rate (year−1; for example 4%/year=0.04year−1)
the net deposit rate (dollars/year).
Here t is the independent variable, P is a constant, and x, I, q are functions of t.
Equations: Now we want to decide how the balance changes as time changes. We’ll estimate the change in the balance Δxas time increases from some time t to a time t+Δt. We can approximate the interest earned per dollar to be:
Note that the units in each of the three terms are dollars/year. Also, there is the initial condition x(0)=P. Thus we have an ODE with initial condition:
Now that the modeling is done, the next step might be to solve this DE, but we won’t do that yet.
Remark 9.2 The notation we chose suggested that the interest rate I depended only on time. However, I could have depended on x as well. This would not change the modeling process. If I does not depend on x, we obtain a linear differential equation. If it does, the equation is nonlinear.
Systems and signals
Let’s get back to the savings account model:
Maybe for financial planning I am interested in testing different saving strategies (different functions q) to see what balances x they result in. To help with this, rewrite the ODE as
In the “systems and signals” language of engineering, q is called the input signal , the bank is the system , and x is the output signal . These terms do not have a mathematical meaning dictated by the DE alone; their interpretation is guided by the system being modeled. But the general picture is this:
The system may be a mechanical system such as an automobile suspension or an electrical circuit, or an economic market. It is impacted by some external signal. We are interested in understanding how the system responds to the external stimulus.
- The input signal is the external stimulus. It usually does not appear in as simple a way in the DE as it does in the example above. But it does always determine the right hand side of the DE (when written in standard linear form).
- The system response (also called output signal ) is the measurable behavior of the system that we are interested in. It is always the unknown function that we write a differential equation for.
- All differential equations have many solutions. The solution of interest is often determined by the state of the system at the beginning. This initial state is given by the initial conditions.
Let’s try to put this into the input/ system response paradigm we’ve just introduced. The system response is the displacement of the mass. This is what we are interested in.
What is the input signal? You could imagine that there are other forces acting on the mass, like there is a sail on the mass, and wind is blowing on the sail creating an input signal. But we are going to start by considering the case where the input signal is 0. Note that pulling the cart back and releasing it specifies the initial state of the system, that is, it gives the initial conditions.
Now we are ready to write down the differential equation . The equation is governed by Newton’s second law
We need to identify the forces acting on the mass. There is the force due to the spring. For the moment, we assume that air resistance is negligible, and there is no friction on the cart.
What is the spring force? When the displacement is positive, the spring is stretched, the force is negative. When the displacement is negative, the spring is compressed, the force is positive. Thus this force is modeled linearly by Hooke’s law:
which is a function of the displacement x away from the neutral position x=0. Note that this linear model is only valid for relatively small displacements. If we stretch the spring too far, the spring force won’t obey this linear law anymore.
The position at time t=0 is x(0)=x0 for some positive displacement x0>0. From the problem statement, we assume that we release the cart with zero initial velocity, x˙(0)=0.
Putting this all together, we get:
with initial conditions:
The last step is to write this in standard linear form . We obtain the following differential equation:
Now let’s consider the same mass/spring system as above where we’ve add a sail to the mass.
The mass now experiences an additional external force from the wind. How does this change the model?
Solution: The model is exactly the same. The only difference is that the input signal is no longer zero, rather it is now the external force due to the wind on the sail. This external force Fwind(t) depends on time in some complicated way that we will not try to write down. The differential equation for this system is:
5 step modeling process
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In the example on the previous page, we outlined a 5 step modeling process that we make explicit here.
- Draw a diagram of the system.
- Identify and give symbols for the parameters and variables of the system.
- Decide on the input signal and the system response. Identify any initial conditions.
- Write down a differential equation relating the input signal and the system response.
- Rewrite the equation in standard linear form with initial conditions.
Written by: Larry Francis Obando – Technical Specialist
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.
Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.
Contact: Ecuador (Quito, Guayaquil, Cuenca)
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