INTRODUCTION
A differential equation is an equation that involves one or more derivatives e.g.
d 2 y dy
(1)
  xy 2  cos 5 x
2
dx
dx
d3y
dy
 6  x 2  ln x
3
dx
dx
(2)
v v
 0
s t
(3)
 2u  2u  2u


 x y
x 2 y 2 z 2
(4)
There are two types of differential equations namely
1. Ordinary Differential equations (ODE)
Equation (1) and (2) are ordinary differential equations
2. Partial differential equation (PDE)
Equation (3) and (4) are partial differential equations.
v – Dependent variable and s,t – independent variables, in (3)
u – Dependent variable and x, y, z – independent variables, in (4)
DEFINITIONS
a) Order of a differential equation
The order of a differential equation is the order of the highest ordered derivative which occurs in
the equation e.g.
d 2 y dy
  y  5x
Order 2
dx 2 dx
d3y d2y
 2  y  x4
Order 3
3
dx
dx
b) degree of a differential equation
The degree of the differential equation is the degree of the highest ordered derivative which
occurs in the equation e.g.
4
d 2 y  dy 
    y  2cos x
dx 2  dx 
Degree 2
Equation
d4y d2y

 6y  0
dx 4 dx 2
4
 d 2 y   dy 
x
 2      3y  e
 dx   dx 
d3y
 5ln x
dx3
4
5
5
 d7y 
d3y  d2y 

5

 7
  8y  0
dx3  dx 2 
 dx 
Order
4
Degree
1
2
4
3
1
7
4
CLASSIFICATION OF ORDINARY DIFFERNTIAL EQUATIONS
Ordinary equation are classified into two groups
a) Linear differential equations
A linear differential equation of order n in the dependent variable x is one which can be
expressed in the form
dny
d n1 y
dy
an  x  n  an1  x  n1  ................a1  x   a0  x  y  r  x 
dx
dx
dx
Where an  x   0
and which is such that
i.
Dependent variable y and its derivatives a are in their first degree
ii.
No product of y or its derivatives are present
iii.
No transcendental functions of y and its derivatives occur. e.g.
2
d y
dy
 7  3y  0
(1)
2
dx
dx
d4y
d3y
d2y
x 4  x 2 3  x3 2  x
(2)
dx
dx
dx
NB
Equation (1) is a linear equation with constant coefficient while equation (2) is a linear equation
with variable coefficient
b) Non-linear differential equations
O.D.Es that do not satisfy all the three conditions are said to be non- linear.
Identify if the equation is linear or non – linear.
i.
ii.
iii.
d3y
dy
 5  6 y2  9
3
dx
dx
2
2
d y
 dy 
 7   4y  0
dx 2
 dx 
d2y
dy
 8y  3y  0
2
dx
dx
ORIGIN OF DIFFERENTIAL EQUATIONS
Differential equations occur in connection to numerous problems that one encounters in various
branches of science and engineering. i. e. differential equations may arise from
a) Geometric problems
b) Physical problems
c) Primitive.
Examples
i.
A falling stone from the top of a building if x is time and y is displacement then
acc  y
Hence we can write a model
y  g
By integrating we get y  gx  v0
By integrating once more we get the displacement as
y   gx  v0 dx
1 2
gx  v0 x  y0
2
Where v0 is the initial velocity and y0 is the displacement when t = 0
ii.
Determination of motion of a projectile. e.g. rocket
iii.
Determination of the charge or current in an electric circuit
If charge q  t  and charge i  t  at time t are related by

i
dq
dt
di
0
dt
Where E is voltage, L inductance and R resistor
The general solution is
E  iR  L
i t  
iv.
v.
vi.
 Rt
E
 Ke L
R
Determining of the vibration of a wire or a membrane
Heat condition in a rod or in a slab
Population growth in human, animal and bacteria e.t.c
The mathematical formulation of such problems gives rise to different equations
Primitive
A primitive is a relation between variables which involve n – essential arbitrary constant e.g.
y  x3  ax
y  Ax 4  Bx
NB: Arbitrary constant are said to be essential if they cannot be replaced by smaller numbers of
constants for example given
y  Ax 4  Bx  Cx
A, B and C are not all essential constants because they can be reduced to 2 constants
y  Ax 4  Bx  Cx
 Ax 4   B  C  x
 Ax 4  Kx
Where k = B+C and therefore there are only 2 essential constants
A primitive involving n-essential arbitrary constant gives rise to differential equations of order n
free of arbitrary constant.
We obtain equation by eliminating the n contest by differentiating n times
Example 1
Obtain the differential equation associated with y  ae2x  be x  ce x
Solution
........................................................ (i)
y  ae2x  be x  ce x
y  2ae2 x  be x  ce x ........................................................ (ii)
y  4ae2 x  be x  ce x ........................................................ (iii)
y  8ae2 x  be x  ce x ........................................................ (iv)
Subtracting (i) from (iii) we get
y  y  3ae2 x
Adding (ii) from (iv) we get
y  y  6ae2 x

y  y  2  y  y 
 y  2 y  y  2 y  0
Example 2
Obtain the differential equation associated with y  a1e3 x  a2e2 x  a3e x
Solution
y  3a1e3 x  2a2e2 x  a3e x
y  9a1e  4a2e  a3e
3x
2x
........................................................ (i)
x
y  27a1e3 x  8a2e2 x  a3e x
........................................................ (ii)
........................................................ (iii)
Adding (i) from (ii) we get
y  y  12 Ae2 x
Adding (ii) from (iii) we get
y  y  36 Ae2 x
y  y  3  y  y 
 y  2 y  3 y  0

Example 3
Obtain the differential equation associated with y  Ae3x  Be x  C
Solution
y  3 Ae3 x  Be x ........................................................ (i)
y  9 Ae3 x  Be x ........................................................ (ii)
y  27 Ae3 x  Be x ........................................................ (iii)
Adding (i) from (ii) we get
y  y  12 Ae2 x
Adding (ii) from (iii) we get
y  y  36 Ae2 x
y  y  3  y  y 
 y  2 y  3 y  0

Example 4
Obtain the differential equation associated with y  Ax3  Bx 2  C
Solution
y  3 Ax 2  2 Bx
y  6 Ax  2 B
y  6 A
A
1
y
6
1
2 B  6 Ax  y  6 yx  y
6
 xy  y
1
 y  3 Ax 2  2 Bx  3  yx 2   xy  y   x
6
1 2
 x y  x 2 y  xy
2
1
 xy  x 2 y
2
1 2
x y  xy  y  0
2
Example 5
Obtain the differential equation associated with x2 y3  x3 y5  0
[implicit function]
dy
dy
 3x 2 y 5  5 x3 y 4
0
dx
dx
dy
2 xy 3  3x 2 y 5   3x 2 y 2  5 x 3 y 4   0
dx
2 xy 3  3x 2 y 2
NB: The equation is a first order since there is only one essential arbitrary constant we can also
write the equation as
 2xy3  3x2 y5  dx  3x2 y 2  5x3 y 4  dy  0
Example 6
Show that a function defined by f  x   a1e2 x  a2e x where a1 and a2 arbitrary constants is
associated with a differential equation of the form y  y  2 y  0
Exercise:
1. Obtain a differential equation associated with the primitive
a) y  a1e3 x  a2e x  a3e x
b) y  Ax 2
c) x 2  9 y 2  B
2. Show that a function defined by f  x   b1e3 x  b2e x where b1 and b2 are arbitrary constants
associated with the differential equation of the form
d2y
dy
 2  3y  0
2
dx
dx
3. Show that a function defined by g  x   c1e3 x  c2 xe3 x  c3e3 x where c1 , c2 and c3 are arbitrary
constants associated with the differential equation of the form
d3y
d2y
dy

3
 9  27 y  0
3
2
dx
dx
dx
4. Show that a function defined by f ( x)  c1e 2 x  c 2 e 2 x  c3 e 4 x where c1 , c2 and c3 are
arbitrary constants associated with the differential equation of the form
d3y
d2y
dy

4
4
 16 y  0
3
2
dx
dx
dx
SOLUTION OF DIFFERENTIAL EQUATIONS
A solution of a differential equation is any function satisfying the equation .In other words the
solution of a differential equation is the primitive associated with it.
Types of Solutions
General solution
This is the primitive associated with the differential equation. The number of arbitrary constant
in the solution is equal to the order of the differential equation. It is the primitive
Particular solution
This is obtained from the general solution by giving numerical values of the arbitrary constant
e.g.
d2y
 y0
y  A sin x  B cos x is a general solution of
dx 2
 2   2 then we would get the particular
If one is given initial condition, say y  0   3 and y 
solution as y  2sin x  3cos x
Singular Solution
Addition solution which cannot be obtained from the general solution (e g primitive)
Exercise
Verify that the given function is a solution of the corresponding differential equations and
determine value of c using given initial conditions
a) y  y  1
y  Ce x  1
y  0   2.5
b) xy  2 y
y  Cx 2
y  2   12
c) yy  x
y 2  x2  C
y  0  1