DIFFERENTIAL EQUATIONS 379
He who seeks for methods without having a definite problem in mind
seeks for the most part in vain. – D. HILBERT
9.1 Introduction
In Class XI and in Chapter 5 of the present book, we
discussed how to differentiate a given function f with respect
to an independent variable, i.e., how to find f ′(x) for a given
function f at each x in its domain of definition. Further, in
the chapter on Integral Calculus, we discussed how to find
a function f whose derivative is the function g, which may
also be formulated as follows:
For a given function g, find a function f such that
dy
dx
= g(x), where y = f (x) ... (1)
An equation of the form (1) is known as a differential
equation. A formal definition will be given later.
These equations arise in a variety of applications, may it be in Physics, Chemistry,
Biology, Anthropology, Geology, Economics etc. Hence, an indepth study of differential
equations has assumed prime importance in all modern scientific investigations.
In this chapter, we will study some basic concepts related to differential equation,
general and particular solutions of a differential equation, formation of differential
equations, some methods to solve a first order - first degree differential equation and
some applications of differential equations in different areas.
9.2 Basic Concepts
We are already familiar with the equations of the type:
x
2
– 3x + 3 = 0 ... (1)
sin x + cos x = 0 ... (2)
x + y = 7 ... (3)
Chapter
9
DIFFERENTIAL EQUATIONS
Henri Poincare
(1854-1912 )
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MATHEMATICS380
Let us consider the equation:
dy
xy
dx
+
= 0 ... (4)
We see that equations (1), (2) and (3) involve independent and/or dependent variable
(variables) only but equation (4) involves variables as well as derivative of the dependent
variable y with respect to the independent variable x. Such an equation is called a
differential equation.
In general, an equation involving derivative (derivatives) of the dependent variable
with respect to independent variable (variables) is called a differential equation.
A differential equation involving derivatives of the dependent variable with respect
to only one independent variable is called an ordinary differential equation, e.g.,
3
2
2
2
d y dy
dx
dx
+
= 0 is an ordinary differential equation .... (5)
Of course, there are differential equations involving derivatives with respect to
more than one independent variables, called partial differential equations but at this
stage we shall confine ourselves to the study of ordinary differential equations only.
Now onward, we will use the term ‘differential equation’ for ‘ordinary differential
equation’.
Note
1. We shall prefer to use the following notations for derivatives:
2 3
2 3
,,
dy d y d y
y yy
dx
dx dx
′ ′′ ′′′
= ==
2. For derivatives of higher order, it will be inconvenient to use so many dashes
as supersuffix therefore, we use the notation y
n
for nth order derivative
n
n
dy
dx
.
9.2.1. Order of a differential equation
Order of a differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the given
differential equation.
Consider the following differential equations:
dy
dx
= e
x
... (6)
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DIFFERENTIAL EQUATIONS 381
2
2
dy
y
dx
+
= 0 ... (7)
3
3 2
2
3 2
dy dy
x
dx dx
+
= 0 ... (8)
The equations (6), (7) and (8) involve the highest derivative of first, second and
third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively.
9.2.2 Degree of a differential equation
To study the degree of a differential equation, the key point is that the differential
equation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc. Consider the
following differential equations:
2
32
32
2
d y d y dy
y
dx
dx dx
+ −+
= 0 ... (9)
2
2
sin
dy dy
y
dx dx
+−
= 0 ... (10)
sin
dy dy
dx dx
+
= 0 ... (11)
We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10)
is a polynomial equation in y′ (not a polynomial in y though). Degree of such differential
equations can be defined. But equation (11) is not a polynomial equation in y′ and
degree of such a differential equation can not be defined.
By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order
derivative involved in the given differential equation.
In view of the above definition, one may observe that differential equations (6), (7),
(8) and (9) each are of degree one, equation (10) is of degree two while the degree of
differential equation (11) is not defined.
Note Order and degree (if defined) of a differential equation are always
positive integers.
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MATHEMATICS382
Example 1 Find the order and degree, if defined, of each of the following differential
equations:
(i)
cos 0
dy
x
dx
−=
(ii)
2
2
2
0
d y dy dy
xy x y
dx dx
dx
+ −=
(iii)
2
0
y
yye
′
′′′
++=
Solution
(i) The highest order derivative present in the differential equation is
dy
dx
, so its
order is one. It is a polynomial equation in y′ and the highest power raised to
dy
dx
is one, so its degree is one.
(ii) The highest order derivative present in the given differential equation is
2
2
dy
dx
, so
its order is two. It is a polynomial equation in
2
2
dy
dx
and
dy
dx
and the highest
power raised to
2
2
dy
dx
is one, so its degree is one.
(iii) The highest order derivative present in the differential equation is
y
′′′
, so its
order is three. The given differential equation is not a polynomial equation in its
derivatives and so its degree is not defined.
EXERCISE 9.1
Determine order and degree (if defined) of differential equations given in Exercises
1 to 10.
1.
4
4
sin( ) 0
dy
y
dx
′′′
+=
2. y′ + 5y = 0 3.
4
2
2
30
ds d s
s
dt
dt
+=
4.
2
2
2
cos 0
d y dy
dx
dx
+=
5.
2
2
cos3 sin 3
dy
xx
dx
=+
6.
2
()
y
′′′
+ (y″)
3
+ (y′)
4
+ y
5
= 0 7.
y
′′′
+ 2y″ + y′ = 0
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DIFFERENTIAL EQUATIONS 383
8. y′ + y = e
x
9. y″ + (y′)
2
+ 2y = 0 10. y″ + 2y′ + sin y = 0
11. The degree of the differential equation
3
2
2
2
sin 1 0
d y dy dy
dx dx
dx
+ + +=
is
(A) 3 (B) 2 (C) 1 (D) not defined
12. The order of the differential equation
2
2
2
230
d y dy
xy
dx
dx
− +=
is
(A) 2 (B) 1 (C) 0 (D) not defined
9.3. General and Particular Solutions of a Differential Equation
In earlier Classes, we have solved the equations of the type:
x
2
+ 1 = 0 ... (1)
sin
2
x – cos x = 0 ... (2)
Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the
given equation i.e., when that number is substituted for the unknown x in the given
equation, L.H.S. becomes equal to the R.H.S..
Now consider the differential equation
2
2
0
dy
y
dx
+=
... (3)
In contrast to the first two equations, the solution of this differential equation is a
function φ that will satisfy it i.e., when the function φ is substituted for the unknown y
(dependent variable) in the given differential equation, L.H.S. becomes equal to R.H.S..
The curve y = φ (x) is called the solution curve (integral curve) of the given
differential equation. Consider the function given by
y = φ (x) = a sin (x + b), ... (4)
where a, b ∈ R. When this function and its derivative are substituted in equation (3),
L.H.S. = R.H.S.. So it is a solution of the differential equation (3).
Let a and b be given some particular values say a = 2 and
4
b
π
=
, then we get a
function y = φ
1
(x) =
2sin
4
x
π
+
... (5)
When this function and its derivative are substituted in equation (3) again
L.H.S. = R.H.S.. Therefore φ
1
is also a solution of equation (3).
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Function φ consists of two arbitrary constants (parameters) a, b and it is called
general solution of the given differential equation. Whereas function φ
1
contains no
arbitrary constants but only the particular values of the parameters a and b and hence
is called a particular solution of the given differential equation.
The solution which contains arbitrary constants is called the general solution
(primitive) of the differential equation.
The solution free from arbitrary constants i.e., the solution obtained from the general
solution by giving particular values to the arbitrary constants is called a particular
solution of the differential equation.
Example 2 Verify that the function y = e
– 3x
is a solution of the differential equation
2
2
60
d y dy
y
dx
dx
+−=
Solution Given function is y = e
– 3x
. Differentiating both sides of equation with respect
to x , we get
3
3
x
dy
e
dx
−
=−
... (1)
Now, differentiating (1) with respect to x, we have
2
2
dy
dx
= 9 e
– 3x
Substituting the values of
2
2
,
d y dy
dx
dx
and y in the given differential equation, we get
L.H.S. = 9 e
– 3x
+ (–3e
– 3x
) – 6.e
– 3x
= 9 e
– 3x
– 9 e
– 3x
= 0 = R.H.S..
Therefore, the given function is a solution of the given differential equation.
Example 3 Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution
of the differential equation
2
2
0
dy
y
dx
+=
Solution The given function is
y = a cos x + b sin x ... (1)
Differentiating both sides of equation (1) with respect to x, successively, we get
dy
dx
= – a sin x + b cos x
2
2
dy
dx
= – a cos x – b sin x
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DIFFERENTIAL EQUATIONS 385
Substituting the values of
2
2
dy
dx
and y in the given differential equation, we get
L.H.S. = (– a cos x – b sin x) + (a cos x + b sin x) = 0 = R.H.S.
Therefore, the given function is a solution of the given differential equation.
EXERCISE 9.2
In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a
solution of the corresponding differential equation:
1. y = e
x
+ 1 : y″ – y′ = 0
2. y = x
2
+ 2x + C : y′ – 2x – 2 = 0
3. y = cos x + C : y′ + sin x = 0
4. y =
2
1
x
+
: y′ =
2
1
xy
x
+
5. y = Ax : xy′ = y (x ≠ 0)
6. y = x sin x : xy′ = y + x
22
xy
−
(x ≠ 0 and x > y or x < – y)
7. xy = log y + C : y′ =
2
1
y
xy
−
(xy ≠ 1)
8. y – cos y = x : (y sin y + cos y + x) y′ = y
9. x + y = tan
–1
y : y
2
y′ + y
2
+ 1 = 0
10. y =
22
ax
−
x ∈ (–a, a) : x + y
dy
dx
= 0 (y ≠ 0)
11. The number of arbitrary constants in the general solution of a differential equation
of fourth order are:
(A) 0 (B) 2 (C) 3 (D) 4
12. The number of arbitrary constants in the particular solution of a differential equation
of third order are:
(A) 3 (B) 2 (C) 1 (D) 0
9.4 Formation of a Differential Equation whose General Solution is given
We know that the equation
x
2
+ y
2
+ 2x – 4y + 4 = 0 ... (1)
represents a circle having centre at (–1, 2) and radius 1 unit.
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MATHEMATICS386
Differentiating equation (1) with respect to x, we get
dy
dx
=
1
2
x
y
+
−
(y ≠ 2) ... (2)
which is a differential equation. You will find later on [See (example 9 section 9.5.1.)]
that this equation represents the family of circles and one member of the family is the
circle given in equation (1).
Let us consider the equation
x
2
+ y
2
= r
2
... (3)
By giving different values to r, we get different members of the family e.g.
x
2
+ y
2
= 1, x
2
+ y
2
= 4, x
2
+ y
2
= 9 etc. (see Fig 9.1).
Thus, equation (3) represents a family of concentric
circles centered at the origin and having different radii.
We are interested in finding a differential equation
that is satisfied by each member of the family. The
differential equation must be free from r because r is
different for different members of the family. This
equation is obtained by differentiating equation (3) with
respect to x, i.e.,
2x + 2y
dy
dx
= 0 or x + y
dy
dx
= 0 ... (4)
which represents the family of concentric circles given by equation (3).
Again, let us consider the equation
y = mx + c ... (5)
By giving different values to the parameters m and c, we get different members of
the family, e.g.,
y = x (m = 1, c = 0)
y =
3
x
(m =
3
, c = 0)
y = x + 1 (m = 1, c = 1)
y = – x (m = – 1, c = 0)
y = – x – 1 (m = – 1, c = – 1) etc. ( see Fig 9.2).
Thus, equation (5) represents the family of straight lines, where m, c are parameters.
We are now interested in finding a differential equation that is satisfied by each
member of the family. Further, the equation must be free from m and c because m and
Fig 9.1
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