differential equation

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 dy dy
= 6 (b) = 3x (c)
dx dx dx = x + 2
2. Sketch the family of curves given by the equation
dy = 2x + 3 and determine the equation
dx
of one of these curves which passes through
the point (1, 3). [y = x 2 + 3x − 1]
46.2 Differential equations
A differential equation is one that contains differential
coefficients.
Examples include
(i) dy
dx = 7x and (ii) d2 y
dx 2 + 5 dy
dx + 2y = 0
Differential equations are classified according to the
highest derivative which occurs in them. Thus example
(i) above is a first order differential equation,
and example (ii) is a second order differential
equation.
The degree of a differential equation is that of the
highest power of the highest differential which the
equation contains after simplification.
( d 2 ) 3 ( )
x dx 5
Thus
dt 2 + 2 = 7 is a second order
dt
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 original
function. This process is called ‘solving the
differential equation’. A solution to a differential
equation which contains one or more arbitrary constants
of integration is called the general solution
of the differential equation.
When additional information is given so that constants
may be calculated the particular solution of
the differential equation is obtained. The additional
information is called boundary conditions. Itwas
shown in Section 46.1 that y = 3x + c is the general
solution of the differential equation dy
dx = 3.
Given the boundary conditions x = 1 and y = 2,
produces the particular solution of y = 3x − 1.
Equations which can be written in the form
dy dy
dy
= f (x), = f (y) and = f (x) · f (y)
dx dx dx
can all be solved by integration. In each case it is
possible to separate the y’s to one side of the equation
and the x’s to the other. Solving such equations
is therefore known as solution by separation of
variables.
46.3 The solution of equations of the
form dy
dx = f (x)
A differential equation of the form dy
dx
solved by direct integration,
∫
i.e. y =
f (x) dx
= f (x) is
Problem 2. Determine the general solution of
x dy
dx = 2 − 4x3
Rearranging x dy
dx = 2 − 4x3 gives:
dy
dx = 2 − 4x3 = 2 x x − 4x3 = 2 x x − 4x2
Integrating both sides gives:
∫ ( ) 2
y =
x − 4x2 dx
i.e. y = 2lnx − 4 3 x3 + c,
which is the general solution.
Problem 3. Find the particular solution of the
differential equation 5 dy + 2x = 3, given the
dx
boundary conditions y = 1 2 when x = 2.
5