A short introduction of Ordinary differential equation

4.  Definition-
A differential equation is an equation containing an unknown
function and its derivatives.
Differential equations are called partial differential equations (PDE) or
ordinary differential equations (ODE) according to whether or not they
contain partial derivatives. The order of a differential equation is the
highest order derivative occurring.
Form of the differential equation-
dy
dx
where P and Q are functions of X or constants , is called the linear
differential equation of the first order . Here X is independent variable
And Y is dependant variable .
— + py = Q

5. Examples:-
32  x
dx
dy
032
2
 ay
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
y is dependent variable and x is independent variable,
and these are ordinary differential equations
1.
2.
3.
Ordinarydifferentialequations

6. The order of the differential equation is order of the
highest derivative in the differential equation.
Differential Equation ORDER
32  x
dx
dy
0932
2
 y
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
1
2
3

7. The degree of a differential equation is power of the
highest order derivative term in the differential equation.
Differential Equation Degree
032
2
 ay
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
03
53
2
2












dx
dy
dx
yd
1
1
3

8. A differential equation is linear, if
1. dependent variable and its derivatives are of degree one,
2. coefficients of a term does not depend upon dependent
variable.
Example:
36
4
3
3






 y
dx
dy
dx
yd
is non - linear because in 2nd term is not of degree
one.
.0932
2
 y
dx
dy
dx
ydExample:
is linear.
1.
2
.

9. Example:
3
2
2
2
x
dx
dy
y
dx
yd
x 
is non - linear because in 2nd term coefficient depends on y.
3.
Example:
is non - linear because
y
dx
dy
sin

!3
sin
3
y
yy is non – linear
4.

10. nth – order linear differential equation
1. nth – order linear differential equation with constant coefficients.
2. nth – order linear differential equation with variable coefficients
If , , ,…. ……… are all constants and Q in some function
of x, then the equation is a linear differential equation with
constant coefficients.

11. WHAT IS HOMOGENEOUS EQUATION
An equation of the form-
Where P
1, P2 ,………. Pn are constants and X is a
function of X , is called the homogeneous
linear equation.

13. 1. Free falling stone -
g
dt
sd
2
2
where s is distance or height and
g is acceleration due to gravity.
2. Spring vertical displacement -
ky
dt
yd
m 2
2
where y is displacement, m is
mass and k is spring constant

14.  Consider the
configuration shown
in Figure in which a
mass m is attached
to one end of a rigid,
but weightless, rod
of length L.
3. The Linear zed Pendulum

15.  The differential equation that describes the motion of
the pendulum is derived by applying Newton’s second
law ma = Fnet along the line tangent to the path of
motion,
or
where γ = c/mL and ω2 = g/L. Due to the presence of the
term sin θ, Eq. cannot be written in the form of a linear
equation. Thus Eq. is nonlinear.
It can be linear zed to
for small θ.

16.  A third example of a second order linear differential
equation with constant coefficients is a model of flow of
electric current in the simple series circuit.
 The current i, measured in amperes, is a function of
time t. The resistance R (ohms), the capacitance C
(farads), and the inductance L (henrys) are all positive
and are assumed to be known constants. The impressed
voltage e (volts) is a given function of time. Another
physical quantity that enters the discussion is the total
charge q (coulombs) on the capacitor at time t. The
relation between charge q and current i is i = dq/dt.
By Kirchhoff’s law,
Eq
cdt
dq
R
dt
qd
L 
1
2
2
4. The Series RLC Circuit

17. q is charge on capacitor,
L is inductance,
c is capacitance.
R is resistance and
e is voltage

18. 5.Newton’s Low of Cooling-
 sTT
dt
dT
 
where
dt
dT
is rate of cooling of the liquid,
sTT  is temperature difference between the liquid ‘T’
and its surrounding Ts

19. dy
y
dt

6. Growth and Decay -
y is the quantity present at any time