prog 24 first-order differential equations

Worked examples and exercises are in the textSTROUD

PROGRAMME 24

FIRST-ORDER DIFFERENTIAL

EQUATIONS


Programme 24: First-order differential equations

Introduction

Formation of differential equations

Solution of differential equations

Bernoulli’s equation



Introduction

A differential equation is a relationship between an independent variable x, a dependent variable y and one or more derivatives of y with respect to x.

The order of a differential equation is given by the highest derivative involved.

2

22

2

34

3

0 is an equation of the 1st order

sin 0 is an equation of the 2nd order

0 is an equation of the 3rd orderx

dyx ydx

d yxy y xdx

d y dyy edx dx



Introduction







Differential equations may be formed from a consideration of the physical problems to which they refer. Mathematically, they can occur when arbitrary constants are eliminated from a given function. For example, let:

2

2

2

2

sin cos so that cos sin therefore

sin cos

That is 0

dyy A x B x A x B xdx

d y A x B x ydx

d y ydx




Here the given function had two arbitrary constants:

and the end result was a second order differential equation:

In general an nth order differential equation will result from consideration of a function with n arbitrary constants.

sin cosy A x B x

2

2 0d y ydx



Introduction







Introduction

Direct integration

Separating the variables

Homogeneous equations – by substituting y = vx

Linear equations – use of integrating factor




Introduction

Solving a differential equation is the reverse process to the one just considered. To solve a differential equation a function has to be found for which the equation holds true.

The solution will contain a number of arbitrary constants – the number equalling the order of the differential equation.

In this Programme, first-order differential equations are considered.




Direct integration

If the differential equation to be solved can be arranged in the form:

the solution can be found by direct integration. That is:

( )dy f xdx

( )y f x dx




Direct integration

For example:

so that:

This is the general solution (or primitive) of the differential equation. If a value of y is given for a specific value of x then a value for C can be found. This would then be a particular solution of the differential equation.

23 6 5dy x xdx

2

3 2

(3 6 5)

3 5

y x x dx

x x x C





If a differential equation is of the form:

Then, after some manipulation, the solution can be found by direct integration.

( )( )

dy f xdx F y

( ) ( ) so ( ) ( )F y dy f x dx F y dy f x dx





For example:

so that:

That is:

Finally:

21

dy xdx y

( 1) 2 so ( 1) 2y dy xdx y dy xdx

2 21 2y y C x C

2 2y y x C





In a homogeneous differential equation the total degree in x and y for the terms involved is the same.

For example, in the differential equation:

the terms in x and y are both of degree 1.

To solve this equation requires a change of variable using the equation:

32

dy x ydx x

( )y v x x





To solve:

let

to yield:

That is:

which can now be solved using the separation of variables method.

32

dy x ydx x

( )y v x x

3 1 3 and 2 2

dy dv x y vv xdx dx x

12

dv vxdx





Consider the equation:

Multiply both sides by e5x to give:

then:

That is:

25 xdy y edx

5 5 5 2 5 75 that is x x x x x xdy de e y e e ye edx dx

5 7 5 7 so that x x x xd ye e dx ye e C

2 5x xy e Ce





The multiplicative factor e5x that permits the equation to be solved is called the integrating factor and the method of solution applies to equations of the form:

The solution is then given as:

where is the integrating factorPdxdy Py Q e

dx

.IF .IF where IFPdx

y Q dx e



Introduction







A Bernoulli equation is a differential equation of the form:

This is solved by:

(a) Divide both sides by yn to give:

(b) Let z = y1−n so that:

ndy Py Qydx

1n ndyy Py Qdx

(1 ) ndz dyn ydx dx




Substitution yields:

then:

becomes:

Which can be solved using the integrating factor method.

1(1 ) (1 )n ndyn y Py n Qdx

(1 ) ndz dyn ydx dx

1 1dz Pz Qdx


Learning outcomes

Recognize the order of a differential equation

Appreciate that a differential equation of order n can be derived from a function containing n arbitrary constants

Solve certain first-order differential equations by direct integration

Solve certain first-order differential equations by separating the variables

Solve certain first-order homogeneous differential equations by an appropriate substitution

Solve certain first-order differential equations by using an integrating factor

Solve Bernoulli’s equation.


prog 24 first-order differential equations

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