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
- 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
- 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