Modeling with differential equations One of the most important application of calculus is differential equations, which often arise in describing some.

Modeling with differential equations

One of the most important application of calculus is differential equations, which often arise in describing some phenomenon in engineering, physical science and social science as well.

Concepts of differential equations In general, a differential equation is an equation that

contains an unknown function and its derivatives. The order

of a differential equation is the order of the highest derivative

that occurs in the equation. A function y=f(x) is called a solution of a differential

equation if the equation is satisfied when y=f(x) and its

derivatives are substituted into the equation.

Example Ex. Show that every member of the family of functions

where c is an arbitrary constant, is a solution of

Sol.

1,

1

t

t

cey

ce

21

( 1).2

y y

2 2

(1 ) (1 )( ) 2

(1 ) (1 )

t t t t t

t t

ce ce ce ce cey

ce ce

22

2 2

1 1 (1 ) 2( 1) 1

2 2 (1 ) (1 )

t t

t t

ce cey

ce ce

21( 1).

2y y

Concepts of differential equations If no additional conditions, the solution of a differential

equation always contains some constants. The solution family

that contains arbitrary constants is called the general solution. In real applications, some additional conditions are

imposed to uniquely determine the solution. The conditions

are often taken the form that is, giving the value

of the unknown function at the end point. This kind of

condition is called an initial condition, and the problem of

finding a solution that satisfies the initial condition is called

an initial-value problem.

0 0( ) ,y t y

Geometric point of view Geometrically, the general solution is a family of solution

curves, which are called integral curves. When we impose an initial condition, we look at the

family of solution curves and pick the one that passes through the point

Physically, this corresponds to measuring the state of a system at time and using the solution of the initial-value problem to predict the future behavior of the system.

0 0( , ).t y

0t

Example

Ex. Solve the initial-value problem

Sol. Since the general solution is substituting

the values t=0 and y=2, we have

So the solution of the initial-value problem is13

13

1 3.

1 3

t t

t t

e ey

e e

21( 1)

.2(0) 2

y y

y

1,

1

t

t

cey

ce

0

0

1 1 12 .

1 1 3

ce cc

ce c

Graphical approach: direction fields For most differential equations, it is impossible to find an

explicit formula for the solution. Suppose we are asked to sketch the graph of the solution of

the initial-value problem The equation tells us that the slope at any point (x,y) on the

graph is f(x,y). To sketch the solution curve, we draw short line segments

with slope f(x,y) at a number of points (x,y). The result is called a direction field.

0 0( , ), ( ) .y f x y y x y

Example Ex. Draw a direction for the equation What

can you say about the limiting value when Sol.

Remark: equilibrium solution

4 2 .y y .x

Separable equations Not all equations have an explicit formula for a solution.

But some types of equations can be solved explicitly.

Among others, separable equations is one type.

A separable equation is a first-order differential equation in

which the expression for can be factored into the

product of a function of x and a function of y. That is, a

separable equation can be written in the form

/dy dx

( ) ( ).dy

f x g ydx

Solutions of separable equations Thus a separable equation can be written into

that is, the variables x and y are separated! We can then integrate both sides to get:

After we find the indefinite integrals, we get a relationship

between x and y, in which there generally has an arbitrary

constant. So the relationship determines a function y=y(x) and

it is the general solution to the differential equation.

( ) ,( )

dyf x dx

g y

( ) .( )

dyf x dx

g y

Example Ex. Solve the differential equation

Sol. Rewrite the equation into

Integrate both sides

which gives

So the general solution is

2 2(1 ) .y y x

22

.1

dyx dx

y

22

,1

dyx dx

y

31arctan .

3y x C

31tan( ).

3y x C

Example Ex. Solve the differential equation

Sol. Separate variables:

Integrate:

which is the general solution in implicit form. Remark: it is impossible to solve y in terms of x explicitly.

26.

2 cos

xy

y y

2(2 cos ) 6y y dy x dx 2 3sin 2y y x C

Example Ex. Solve the differential equation

Sol.

C is arbitrary, but is not arbitrary. While we can verify

y=0 is also a solution. Therefore

where A is an arbitrary constant, is the general solution.

2 .y x y33

2 3ln | |3

xCdy x

x dx y C y e ey

Ce

3

3

x

y Ae

Orthogonal trajectories An orthogonal trajectory of a family of curve is a curvethat intersects each curve of the family orthogonally. Forinstance, each member of the family of straight linesis an orthogonal trajectory of the family To find orthogonal trajectories of a family of curve, firstfind the slope at any point on the family of curve, which isgenerally a differential equation. At any point on theorthogonal trajectories, the slope must be the negativereciprocal of the aforementioned slope. So the slope oforthogonal trajectories is governed by a differential equation,too. Last solve the equation to get the orthogonal trajectories.

y mx2 2 2.x y r

Example Ex. Find the orthogonal trajectories of the family of curves

where k is an arbitrary constant. Sol. Differentiating we get or

Substituting into it, we find the slope at any point is

At any point on orthogonal trajectory, the slope is

Solving the equation, we get

2 ,x ky2 ,x ky 2 ,dx kydy

1

2

dy

dx ky

2/k x y

.2

dy y

dx x

2.

dy x

dx y

22 .

2

yx C

Example Ex. Suppose f is continuous and

Find f(x). Sol.

2

0( ) ( ) ln 2.

2

x tf x f dt

0( ) 2 ( ) ln 2

xf x f u du

( ) 2 ( ), (0) ln 2f x f x f

2( ) ln 2.xf x e

Homework 21 Section 8.2: 8, 14, 29

Section 8.3: 28, 29

Page 583: 7, 8, 10

Section 9.1: 10, 11