Higher Engineering Mathematics

444 DIFFERENTIAL EQUATIONS

Now try the following exercise.

Exercise 177 Further problems on families

of curves

1. Sketch a family of curves represented by each
   of the following differential equations:

(a)

dy

dx

=6 (b)

dy

dx

= 3 x(c)

dy

dx

=x+ 2

2. Sketch the family of curves given by the equa-

tion

dy

dx

= 2 x+3 and determine the equation

of one of these curves which passes through

the point (1, 3). [y=x^2 + 3 x−1]

46.2 Differential equations

Adifferential equationis one that contains differ-
ential coefficients.

Examples include

(i)

dy

dx

= 7 x and (ii)

d^2 y

dx^2

+ 5

dy

dx

+ 2 y= 0

Differential equations are classified according to the

highest derivative which occurs in them. Thus exam-

ple (i) above is afirst order differential equation,

and example (ii) is asecond order differential

equation.

Thedegreeof a differential equation is that of the

highest power of the highest differential which the

equation contains after simplification.

Thus

(

d^2 x

dt^2

) 3

+ 2

(

dx

dt

) 5

=7 is a second order

differential equation of degree three.

Starting with a differential equation it is possible,

by integration and by being given sufficient data to

determine unknown constants, to obtain the origi-

nal function. This process is called‘solving the

differential equation’. A solution to a differential

equation which contains one or more arbitrary con-

stants of integration is called thegeneral solution

of the differential equation.

When additional information is given so that con-

stants may be calculated theparticular solutionof

the differential equation is obtained. The additional

information is calledboundary conditions.Itwas

shown in Section 46.1 thaty= 3 x+cis the general

solution of the differential equation

dy

dx

= 3.

Given the boundary conditionsx=1 andy=2,

produces the particular solution ofy= 3 x−1.

Equations which can be written in the form

dy

dx

=f(x),

dy

dx

=f(y) and

dy

dx

=f(x)·f(y)

can all be solved by integration. In each case it is

possible to separate they’s to one side of the equa-

tion and thex’s to the other. Solving such equations

is therefore known as solution byseparation of

variables.

46.3 The solution of equations of the

form

dy

dx

=f(x)

A differential equation of the form

dy

dx

= f(x)is

solved by direct integration,

i.e. y=

∫

f(x)dx

Problem 2. Determine the general solution of

x

dy

dx

= 2 − 4 x^3

Rearrangingx

dy

dx

= 2 − 4 x^3 gives:

dy

dx

=

2 − 4 x^3

x

=

2

x

−

4 x^3

x

=

2

x

− 4 x^2

Integrating both sides gives:

y=

∫ (

2

x

− 4 x^2

)

dx

i.e. y=2lnx−

4

3

x^3 +c,

which is the general solution.

Problem 3. Find the particular solution of the

differential equation 5

dy

dx

+ 2 x=3, given the

boundary conditionsy= 1

2

5

whenx= 2.