cm1902 integration 9 - odes

8/12/2019 CM1902 Integration 9 - ODEs

1/11

Integration Lecture 9

1st

Order Differential Equations

CM1902 Mathematics 1B


2/11

Ordinary Differential Equations,

ODEs

There are lots of situations in engineering where wecan measure the rate of change of a variable, e.g.

Velocity or acceleration and wish to find an

expression for the underlying independent variable,

e.g. If we know the force on a body we can useNewtons 3rdlaw to find the acceleration but we want to

know the displacement with respect to time.

The above are examples of an Ordinary Differential

Equation There are lots of these equations which are standard

results in engineering maths. They are specified

according to their order. If the equation involves a

first derivative then it is first order, etc.2011-122


3/11

1stOrder ODEs

These are equations involving both a functionsuch as x(t) and its 1stderivative

There are lots of different families of theseequations which depends on the form of f(x, t).

We will look at 2 special cases:

2011-123

dxf(x,t)

dt

dxf(t)

dt

dxf(x)g(t)

dt

We can just integrate

each side of this equation

with respect to t

This is trickier!


4/11

Examples 1

Solve the 1storder ODE

2011-124

3tdx

6ed t

Solve the 2ndorder ODE 2

2

1 d y1

x dx


5/11

Initial Conditions

To solve a differential equation we must integrateand that introduces an arbitrary constant. That

means that every differential equation has an

infinite number of solutions (an infinite number of

constants).

We usually get a differential equation given along

with a condition that the dependant variable takes a

certain value at a particular value of the

independent variable.

For example, y = 3 when x = 0 or x = 19 when t = 7

Because the condition is often given at x = 0 or t =

0 it is known as an initial condition

The differential e uation is then called an Initial2011-125


6/11

Example 1bSolve the 1storder ODE

subject to the condition x = 3 when t= 0

2011-126

3tdx 6ed

t

3tx 2e +CFrom before we had:

This is known asthe GENERAL

solution

This is known as the PARTICULAR solution


7/11

Separation of Variables

Given an equation of the form

We can solve by thinking of as a ratio of

differentials dx and dt. Then

Or more correctly

Now we have two integrals, one in x and one in t.

The variables are separated, mixtures of x and t on

one side or another are not allowed. 2011-127

dx f(x)g(t)dt

dx

dt

1dx g(t)dt

f(x)

1dx g(t)dt

f(x)


8/11

Example 2Solve the 1storder ODE

subject to the condition x = 1 when t= -1

2011-128

dxx

dt


9/11


subject to the condition x(0) = 1

2011-129

2dx x cos(t)dt


10/11


subject to the condition y(2) = 1

2011-1210

2dy x xydx


11/11

Summary

Differential equations crop up regularly inengineering problems. In years 2 and 3 much of

the mathematics modules are taken up with

obtaining solutions to standard problems

Two methods we have looked at for 1storder

ODEs are

Direct Integration

Separation of Variables

An (initial) condition gives allows us to obtain a

particular solution to an IVP

Now: Tutorial 9 2011-1211

cm1902 integration 9 - odes

Documents