ECON351
Differential Equation
Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler
MITx: 18.031x Introduction to Differential Equations
Mathematics 306 Differential Equations
http://sam-dolan.staff.shef.ac.uk/mas212/
http://sam-dolan.staff.shef.ac.uk/mas212/docs/l4.pdf
http://sam-dolan.staff.shef.ac.uk/mas212/docs/2016_Class_Test_1_answers.html
https://en.wikipedia.org/wiki/Isocline
https://en.wikipedia.org/wiki/Nullcline
Can you guess my secret function y(t)?
Clue: It satisfies the differential equation
which could model population growth in some biological system, like the growth of yeast cells. Note that y˙ is common notation for the derivative with respect to time; y˙, y′, and dy/dt all mean the same thing here.
Maybe you guessed y=e3t. This is a solution to the differential equation above, because substituting it into the DE gives 3e3t=3e3t. But it’s not the function I was thinking of! Some other solutions are y=7e3t, y=−5e3t, y=0, etc. From calculus we know that the family of functions
is the general solution .
Saying this means that
for each number c, the function y=ce3t is a solution, and
there are no other solutions besides these.
So there is a 1-parameter family of solutions to this DE. The constant c is a parameter .
You still haven’t guessed my secret function.
Clue 2: My function satisfies the initial condition y(0)=5.
There is a number c such that y(t)=ce3t holds for all t; we need to find c. Plugging in t=0 shows that
Thus, among the infinitely many solutions to the DE, the solution satisfying the initial condition y(0)=5 is
Definition 3.2 An initial value problem is a differential equation together with initial conditions.
Important: Checking a solution to a DE is usually easier than finding the solution in the first place, so it is often worth doing. Just plug in the function to both sides, and also check that it satisfies the initial condition.
In the solution to the DE, only one initial condition was needed, since only one parameter c needed to be recovered.
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.
The system
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.
y number of cells t 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
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.
Definition 5.1 The order of a DE is the highest n such that the nth derivative of the function appears.
The equation is an ODE because it only involves derivatives with respect to one variable, the variable t.
homogeneous linear ODE
Most general nth order homogeneous linear ODE:
for some functions pn(t), …, p0(t) called the coefficients .
An inhomogeneous linear ODE is the same except that it has also one term that is a function of t only .
Most general nth order inhomogeneous linear ODE:
Either of the two forms above can be reduced further by dividing the entire DE by pn(t), the coefficient of the highest derivative y(n) becomes 1. A differential equation written in either of the two forms above but with leading coefficient 1 is said to be in standard linear form .
Remark 6.2 If you already know that an ODE is linear, there is an easy test to decide if it is homogeneous or not: Plug in the constant function y=0.
If y=0 is a solution, the ODE is homogeneous.
If y=0 is not a solution, the ODE is inhomogeneous.
Remark 6.3 The cell division example gives us a homogeneous equation. If you start with no yeast cells, the population stays at 0.
For an ODE tobe nonlinear, the functions y, y˙, … must enter the equation in a more complicated way: raised to powers, multiplied by each other, or with nonlinear functions applied to them.
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.
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
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.
Guidelines:
1.Identify relevant quantities, both known and unknown, and give them symbols. Find the units for each.
2.Identify the independent variable(s). The other quantities will be functions of them, or constants. Often time is the only independent variable.
3.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.
(Once we have the language of input signals and system response, we’ll refine these guidelines.)
9. Model of a savings account
Simplifying assumptions:
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:
P the initial amount that the account starts with (dollars) t time from the start (years) x balance (dollars) I the interest rate (year−1; for example 4%/year=0.04year−1) q the net deposit rate (dollars/year).
The smaller Δt is, the better the approximation becomes, and in the limit as Δt→0
This yields the differential equation
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.
11. Systems and signals
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:
We’ll see many examples that model phenomena using this input/system response language.
1.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.
2.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).
3.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.
4.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.
1. Solving first-order linear ODEs
Objectives
Find the general solution to any homogeneous first order (linear) ODE using separation of variables .
Find a particular solution to any inhomogeneous first order linear ODE using variation of parameters .
Find the general solution to any inhomogeneous first order differential equation by first finding the solution to the associated homogeneous function, finding one particular solution, and applying superposition .
Solve initial value problems for first order linear ODEs.
2. Separation of variables for first order ODEs
Separation of variables is a technique that reduces the problem of solving certain first-order ODEs to evaluating two integrals.
1.Separation of variables works when we can write the equation in the form
2.Isolate the derivative and express
Separation: Put the terms involving y on the left and terms involving t on the right.
Warning: We divided by y, so at some point we will have to check y=0 as a potential solution.
3.Integrate
Note: The method reduces to evaluating two integrals separately, one in y and one in t. Observe that we combined the two constants C1 and C2 of integration into one constant C. We will combine the two constants to form C=C2−C1 immediately from now on.
4.Solve for y.
As C runs over all real numbers, the coefficient ±eC runs over all nonzero real numbers. Thus the solutions we have found so far are
5.Because we divided by y, there is possibly an exceptional solution y=0. Checking directly, it turns out that it is indeed a solution. It can be considered as the function cet2 for c=0.
6.(Optional) Double check your solution. Plugging in y=cet2 to y˙−2ty=0 gives cet2(2t)−2tcet2=0, which is true, as it should be.
If you are lucky, you can write h(t,y) as a product of two functions, one of which is purely a function of t; and the other purely a function of y:
Note: If there are factors involving both variables, such as y+t, then it is impossible to separate variables; in this case, give up and try a different method.
The technique of multiplying by differentials allows us to work backwards to the formula for the solution.
Homogeneous Second Order Linear Differential Equations
https://www.youtube.com/watch?v=soU-zRdpsoA
Second-Order Non-Homogeneous Differential
https://www.youtube.com/watch?v=h3SCtTtlCKU