23MA202-Mathematical Foundations for Engineering - ODE

23MA202-Mathematical Foundations for
Engineering
Ordinary Differential Equations
By
Dr. S.Shyamala AP/Maths
Ms.G. Nandhini AP/Maths
Excel Engineering College (Autonomous)

Differential Equations
Differential Equations were begun by Newton and
Leibniz in the seventeenth century.
Differential Equations are often used in economics.
A Differential Equation is an equation which involves
derivatives.
It is an implicit functional relationship between variables
and their differentials.
The unknown is often a function of time.

Order and degree of Differential Equations
The order of a Differential Equation is the highest order of
any of the derivatives contained in it.
A Differential Equation is of order n if the nth
derivative is
the highest order derivative in it.
The highest power to which the derivative of the highest
order occur is the degree of a Differential Equation.

Example of Differential Equation
 Y = dy/dt + t2
Differential Equation of
order 1 and degree 1
 (d2
y/dt2
)3
– 6 (dy/dt)2
+ty = 0
Differential Equation of
order 2 and degree 3

Kinds of Differential Equations
 Ordinary Type:
Involving only one independent variable and its
derivatives.
Partial Differential Equations:
Involving more than one independent variable and partial
derivatives.

Solution of Differential Equation
Differential Equation of order n has a General solution
containing n arbitrary independent constants.
A Particular solution may be obtained from the general
solution by assigning certain specific values [which are
obtained from any given information of the initial
conditions given in the problem] to the arbitrary
constants .

First- order linear Differential Equation with constant
coefficient and constant term
General form of the equation:
dy/dt + u(t) y = w(t)
derivative is taken with respect to time (t).
u and w are functions of t.
When u and w are taken as constant, the is termed as
First- order linear Differential Equation with constant
coefficient and constant term.

Homogeneous Case
U = a constant ; a
W = 0
The form of the Equation becomes :
dy/dt + ay = 0 ……… 1
=> 1/y dy/dt = -a
=> ∫ 1/y dy = - ∫ a dt [taking integration of both side]
From left side, ∫ 1/y dy = log │y│ + c1 (y#0)
From right side, - ∫ a dt = -at + c2
By equating, log │y│ = -at + c

Now taking anti log of log│y│implies, elog y
= x
therefore, , log │y│ = -at + c becomes,
elog │y│
= e-at + c
=> │y│= e-at
. ec
= A. e-at
.
Therefore the solution is
y(t) = A. e-at
……………… 2
This is general solution, where A is arbitrary.
For any particular value of ‘A’ the solution becomes “particular.”
There may be infinite number of particular solution.
Setting t=o , y(0) = A . e0
=> A = Y (0)
Therefore Y(t) = Y(0)e-at
…………….. 3
This is the Definite Solution as the arbitrary constant is definitized here.
Y(0) is the only value that makes the solution satisfy the initial condition.

Note:
The solution is a function of Y(t) – a time path where “ t”
symbolizes time.
No derivative is present in the solution.

The Non-homogeneous Case
The equation is
dY/dt + aY = b ....... 4
The solution is the sum of Complementary function and
Particular Integral.
To find Particular integral ( Yp ) , y is taken as a constant.
Here Y = k , dY/dt = 0
Equation 4 becomes, ak = b => k = b/a = Yp ... 5a

To get the Complementary Function (Yc ), the process is to get
the homogeneous equation which is termed as reduced
equation.
The Trial solution is
Yc = A e –at
....5b
So, the general solution becomes
Y(t) = Yc + Yp = A e –at
+ b/a ........6

Now the constant A is definitized by means of an
initial condition.
When t = 0 , y takes the value Y(0) .
Equation 6 becomes,
Y(0) = A + b/a => A = Y(0) - b/a
Therefore the definite solition is
Y(t) = [ Y(0) – b/a] e -at
+ b/a ........7

Use of Differential Equation in Economics
Differential Equations have wide use to explain different
concepts of Economics. Equilibrium, Stability of different
Market models can be easily explained by this
mathematical method.
Growth models also use linear differential Equation.

Analysis of Equilibrium is an important part of Economics.
Differential Equations explains the Euilibrium as well as the investigates the
Stability of Equilibrium.
In the above discussed linear differential equation, the time path of Y is obtained.
The solution is given here once more:
Y(t) = [ Y(0) – b/a] e -at
+ b/a ............ 7
Equilibrium can be attained if , Y(0) = b/a
When Y(0) ≠ b/a , it requires to analyse whether the time path converges
to equilibrium over time or not and this is called the Stability Analysis.
The stability depends on the value of “a”.
From equation 7 it can be shown that, starting from a non-equilibrium position,
Convergence is obtained if e-at
→ 0 as t→ ∞ It is possible only when a > 0
Divergence is obtained e-at
→ ∞ as t→ ∞ It is possible only when a < 0

23MA202-Mathematical Foundations for Engineering - ODE

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