Higher Engineering Mathematics, Sixth Edition

Solution of first order differential equations by separation of variables 445

x

10

20

30

y

y



2 x

2 

15

y 

2 x

2 

8

y 

2 x

2 

6

y



2 x

2

 1

 10

 41  3  2 0 234

Figure 46.2

Now try the following exercise

Exercise 176 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 differential
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 high-

est 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 differ-

ential 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 original

function. This process is called‘solving the differ-

ential equation’. A solution to a differential equation

which contains one or more arbitrary constants of inte-

gration is called thegeneral solutionof the differential

equation.

When additional information is given so that con-

stants may be calculated theparticular solutionof the

differential equation is obtained. The additional infor-

mation is calledboundary conditions.Itwasshownin

Section 46.1 thaty= 3 x+cis the general solution of

the differential equation

dy

dx

= 3.

Given the boundary conditionsx=1andy=2, pro-

duces 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 besolved by integration.In each caseit is possible

to separate they’s to one side of the equation 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