DiffEquations[Unit 1]

Differential Equations (MATH313)

Basic Concepts Page 1

BASIC CONCEPTS

1.1 Differential Equations

A differential equation is an equation involving an unknown function and its derivatives. Differential

equations play an important role in the construction of mathematical models to approximate real-world

problems.

The following are examples of differential equations:

dydx = cos x (1.1)

dydx + ky = 0 (1.2)

x + ydx 2xy dy = 0 (1.3)

ut = h

ux +

uy (1.4)

L didt + Rdidt +

1C i = E cos t (1.5)

Vx +

Vy = 0 (1.6)

dwdx

xy dwdx + w = 0 (1.7)

d xdy + x

dxdy 4xy = 0 (1.8)

dydx + 7 #

dydx$

8y = 0 (1.9)



dydt +

dxdt = x (1.10)

x fx + yfy = nf (1.11)

When an equation involves one or more derivatives with respect to a particular variable, that variable is

called an independent variable. A variable is called dependent if a derivative of that variable occurs.

Parameters are those that are considered constants in the equation for a particular situation but may take

on a range of different values depending on the situation.

Example 1.1. In equation 1.5,

L didt + Rdidt +

1C i = E cos t (1.5)

the variable t is the independent variable, i is the dependent variable. This is because the differential of i which is di is differentiated with respect to t. The variables L, R, C, E and are considered constants in this equation and are called parameters.

Example 1.2. In equation 1.6, Vx +

Vy = 0 (1.6)

the dependent variable is V and the independent variables are x and y.

Example 1.3. In equation 1.3, x + ydx 2xy dy = 0 (1.3) the equation is written implicitly. Therefore, we may consider the variables x to be the independent variable, y the dependent one or vice versa. If the equation is written implicitly, it is written in terms of the variables differentials.



1.2 Definitions

A differential equation is an ordinary differential equation (ODE) if the unknown function depends on only

one independent variable. If the unknown function depends on two or more independent variables, the

differential equation is a partial differential equation (PDE).

Example 1.4. Equations 1.4, 1.6, and 1.11 are partial differential equations while the rest are ordinary differential equations.

The order of a differential equation is the order of the highest-ordered derivative appearing in the equation.

The degree of a differential equation is the power or the exponent - of the highest-ordered derivative

appearing in the equation.

Example 1.5. The equation dydx + 2b #

dydx$

+ y = 0 (1.12) is of order two and of degree one.

Example 1.6. The equation

dydx

+ 3y #dydx$* + y #dydx$

= 5x (1.13) is of order two and of degree three.

The ordinary differential equation

F-x, y, y/, , y1 2 = 0 (1.14)

is said to be linear if F is a linear function of the variables y, y, . . . , y(n) which is also true for partial

differential equations. Thus, the general linear ordinary differential equation of order n is

a4xy1 + a5xy165 + + a1xy = gx (1.15)

This means that for an ordinary differential equation to be linear, all the coefficients of the terms are

functions of the independent variable only. Equations which do not satisfy this condition are considered to

be non-linear.



Example 1.7. The equations

t dydt + tdydt + 2y = sin t (1.16)

d8y

dt8 + d ydt +

dydt +

dydt + y = 1 (1.17)

and d y

dt + tdydt + y cost = t (1.18)

are all linear differential equations whereas the equations

1 + y dydt + tdydt + y = e: (1.19)

dydt + ty = 0 (1.20)

And dy

dt + sint + y = sin t (1.21) are all non-linear differential equations.

Oftentimes, situations can give rise to a non-linear model. This forces us to linearize a model for some

certain restrictions so we can arrive at a solution.

Example 1.8. The motion of an oscillating pendulum is modeled as d

dt + gL sin = 0 (1.22)

Figure 1. An oscillating pendulum



where g = acceleration due to gravity; L = length of the string; = angle of the pendulum from the normal and m = mass of the pendulum bob. The model is obviously a non-linear equation. But take note that for small values of , we have sin , hence the model 1.22 can be approximated by d

dt + gL = 0 (1.23)

which is now a linear equation. Such process is called linearization and greatly simplifies the process of finding a solution.

If one cannot help but solve a non-linear equation, then he may use computer-aided numerical methods to

arrive at a solution.

A function , defined in the interval a < x < b, is called a solution of the differential equation

y1 = f-x, y, y/, , y1652 (1.24)

provided that the n derivatives of the function exist on the interval a < x < b and

1x = f =x, n, / , , 165x> (1.25)

Note that equation 1.24 is equation 1.14 solve explicitly for y(n) in terms of the other n + 1 variables under

suitable restrictions on F.

Example 1.9. Verify that y = e? is a solution of the equation dy

dx + dydx 6y = 0 (1.26)

Example 1.10. Show that the function fx = x + e? is a solution to y// 2y/ = 2 (1.27)



The general solution of a differential equation is the set of all solutions obtained from integrating the

differential equation and contains the arbitrary constants. The order of the differential equation determines

the number of arbitrary constants.

The particular solution of a differential equation is the solution obtained by assigning specific values to the

arbitrary constants in the general solution. This can be done by determining certain conditions that would

lead to the determination of the arbitrary constants. These conditions fall into two categories: the initial

conditions and the boundary conditions.

The initial conditions are subsidiary conditions on the unknown function and its derivatives, all given at the

same value of the independent variable. Such differential equations and the subsidiary conditions constitute

an initial-value problem (IVP).

The boundary conditions are subsidiary conditions given at more than one value of the independent

variable. Such differential equations and the subsidiary conditions constitute a boundary-value problem

(BVP).

Example 1.11. The general solution of the differential equation y// + 4y = 0 (1.28) is y = c5 sin 2x + c cos 2x (1.29) where c1 and c2 are the arbitrary constants. Note that equation 1.26 is a second-order equation, so the solution should contain two arbitrary constants. Try to verify that equation 1.27 is indeed the solution of equation 1.26. If we specify the conditions y(0) = pi and y(0) = 2pi, we have an initial value problem. Using these initial conditions, we find that c1 = c2 = pi. Hence a particular solution for the differential equation 1.26 is y = sin 2x + cos 2x (1.30) On the other hand, if we specify the conditions as y(0) = 2pi and y(pi/4) = 3pi, we have a boundary value problem. Thus using these boundary conditions, we find that c1 = 3pi and c2 = 2pi, and a particular solution to the differential equation 1.26 is y = 3 sin 2x + 2 cos 2x (1.31)

Some important questions about the solutions of the differential equation:

Does equations of the form

y1 = f-x, y, y/, , y1652 (1.24) always have a solution?



If we assume that the equation of 1.24 has at least one solution, how many actual solutions does it

have, and what additional conditions must be specified to single out a particular solution?

Given a differential equation of the form of 1.24, can we actually determine a solution, and if so,

how?

The answers:

For the first question, the answer is NO. Not all differential equations that can be written in the form

of 1.24 do have a solution. So how can we tell whether some particular equation has a solution?

This is the question of existence of a solution, and it is answered by theorems stating that under

certain restrictions on the function f in 1.24, the equation always has solutions.

However, the existence of a solution is not a purely mathematical concern, for at least two reasons:

(1) if a problem has no solution, we would prefer to know that fact before investing time and effort

in a vain attempt to solve the problem; (2) if a sensible physical problem is modeled mathematically

as a differential equation, then the equation should have a solution; if none, then there is

presumably wrong with the formulation.

The second question is about the uniqueness of the solution. In general, solutions of differential

equation contain one or more arbitrary constants of integration (see example 1.11). How many

solutions does a differential equation have? Depending on the nature and restrictions on the

arbitrary constants, the answer is up to infinity, corresponding to the infinite possibilities for the

arbitrary constant/s. If we want a particular solution, then we need certain conditions to specifically

determine values for the arbitrary constant/s. Thus, initial and/or boundary conditions are needed

to be specified.

If the answer to the third question is a YES, then we were able to answer the first question, also

with a YES. But if we do not have any knowledge as to the existence of the solution, then we may

initially use a computer to find a numerical approximation of the solution. However, care must be

taken as the approximate solution found by a computer is actually a solution it may be

practically impossible. Or else, if it is indeed a valid solution, but cannot be expressed in

elementary mathematical functions polynomial, trigonometric, exponential, logarithmic, and

hyperbolic functions then it may be of no practical use to us.



Drill Problems For items 1 thru 19, state whether the equation is ordinary or partial, linear or non-linear, give its order and degree, and name the dependent and independent variables and parameters.

1.

dxdt + kx = 0

2.

wt = a

wx

3. x + ydx + 2xy dy = 0

4. y/ + Pxy = Qx

5. y/// 3y/ + 2y = 0

6. yy// = x

7.

ux +

uy +

uz = 0

8.

d8ydx8 = wx

9.

x dydt ydxdt = c5

10.

L didt + Ri = E

11. x + ydx + 3x 1dy = 0

12. xy + y8 y = 0



13. d wdx

2 #dwdx $

8 + yw = 0

14.

dydx = 1 xy + y

15. y// + 2y/ 8y = x + cos x

16. a da + b db = 0 17.

eF dydx + 2 #dydx$

= 1

18.

4 d ydx + sin xdydx + 5xy = 0

19. yt 4

yx

20. Verify that sin kt is a solution of the equation in drill problem 1. 21. Verify that e-2x is a solution of the equation in drill problem 5. 22. Verify that 3e-3x + 4ex is a solution of the equation in drill exercise 5.

23. Verify that for x > 0,

x / is a solution of the equation of Exercise 6. 24. Determine whether yx = 2e6? + xe6? is a solution of y// + 2y/ + y = 0 25. Is yx = 1 a solution of y// + 2y/ + y = x? 26. Show that y = ln x is a solution of xy// + y/ = 0 for x > 0. 27. Determine whether any of the functions

a. y5x = sin 2x b. yx = x; or c. y x = 5 sin 2x

is a solution to the initial value problem y// + 4y = 0; y(0) = 0, y(0) = 1. 28. Find a solution to the boundary-value problem y// + 4y = 0; y =JK> = 0, y =JL> = 1, if the general

solution to the differential equation is yx = c5 sin 2x + c cos 2x.



1.3 Some Basic Mathematical Models

Before embarking on a serious study of differential equations, you should have some idea of the possible

benefits to be gained by doing so.

Many of the principles, or laws, underlying the behavior of the natural world are statements or relations

involving rates at which things happen. When expressed in mathematical terms, the relations are equations

and the rates are derivatives, thus, these give rise to differential equations.

Some engineering situations involving rates of change:

Motion of fluids

Flow of current in electric circuits

Dissipation of heat in solid objects

Propagation and detection of seismic waves

Increase or decrease of population

In constructing mathematical models, one should recognize that different situations pose different

problems, and that they should be attacked in different approaches. In fact, successful modeling is not a

skill that can be reduced to the observance of a set of prescribed rules. Nevertheless, it may be helpful to

list some steps that are often part of the process:

1. Identify the independent and dependent variables and assign letters to represent them.

2. Choose the units of measurement for each variable. The choice of units is somewhat arbitrary, but

some choices may be much more convenient than others.

3. Articulate the basic principle that underlies or governs the problem you are investigating. This may

be a widely recognized physical law or it may be a more speculative assumption that may be

based on your own experience or observations.

4. Express the principle or law in step 3 in terms of the variables you chose in step 1. It may require

the introduction of physical constants or parameters and the determination of appropriate values

for them.

5. Make sure that your equation is expressed in correct units.

6. Depending on the complexity of the model constructed, special methods, like computer-aided

numerical methods, may be employed to solve the given model.

As an example, we begin with two models leading to equations that are easy to solve.

Example 1.12. Suppose that an object is falling in the atmosphere near sea level. Formulate a differential equation that describes the motion. Then investigate the behavior of the solutions of the obtained differential equation.



Figure 2. Direction field for the model of Example 1.12

Direction fields are valuable tools in studying the solutions of differential equations of the form

dydt = ft, y (1.32)

where f is a given function of the two variables t and y, sometimes referred to as the rate function.

A useful direction field for equations of the general form 1.32 can be constructed by evaluation f at each

point of a rectangular grid consisting of at least a few hundred points. Then, at each point on the grid, a

short line segment is drawn whose slope is the value of f at that point. Thus, each line segment is tangent

to the graph of a solution passing through that point.

A direction field drawn on a fairly fine grid gives a good picture of the overall behavior of solutions of a

differential equation. The construction of a direction field is often a useful first step in the investigation of a

differential equation.

In constructing a direction field, we do not have to solve 1.32 but merely evaluate the rate function f(t, y)

many times. Thus, direction fields can be readily constructed even for equations that may be quite difficult

to solve. Also, repeated evaluation of a given function is a task for which a computer is well suited and you

should usually use a computer to draw a direction field.

Example 1.13. Consider a population of field mice who inhabit a certain rural area. In the absence of predators, we assume that the mouse population increases at a rate proportional to the current population. Write down a mathematical for this. Suppose that several owls live in the same neighborhood and that they kill n mice per day. Modify the first model to account for this. Then investigate the model using a direction field.



Figure 3. A direction field for the equation obtained in Example 1.13

Drill Problems

1. A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of 300 gal/min. The mixture flows out at the same rate so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond,

a. Write a differential equation whose solution is the amount of chemical in the pond at any time.

b. How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially?

2. A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.

3. A certain drug is being administered intravenously to a hospital patient. Fluid containing 5 mg/cm3 of the drug enters the patients bloodstream at a rate of 100 cm3/hr. The drug is absorbed by body tissues or otherwise leaves the blood stream at a rate proportional to the amount present, with a rate constant of 0.4/hr.

a. Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the amount of drug that is present in the bloodstream at any time.

b. How much of the drug is present in the bloodstream after a long time?

DiffEquations[Unit 1]

Documents

Transcript of DiffEquations[Unit 1]