It describes about the application of ordinary equations.

2. Differential Equations
A differential equation is an equation which contains the derivatives of a
variable, such as the equation
Here x is the variable and the derivatives are with respect to a second
variable t. The letters a, b, c and d are taken to be constants here. This
equation would be described as a second order, lineardifferential
equation with constant coefficients. It is second order because of the
highest order derivative present, linear because none of the derivatives
are raised to a power, and the multipliers of the derivatives are constant.
If x were the position of an object and t the time, then the first derivative
is the velocity, the second the acceleration, and this would be an equation
describing the motion of the object. As shown, this is also said to be
a non-homogeneous equation, and in solving physical problems, one
must also consider the homogeneous equation.

3. Applications of First-order Differential
Equations to Real World Systems
1.Cooling/Warming Law
2.Population Growth and Decay
3.Radio-Active Decay and Carbon Dating
4.Mixture of Two Salt Solutions
5.Series Circuits

4. Application of differential
equations to electric circuits
•Resistive circuits: it is an electrical circuit in
which a resistor and a source of electricity are
connected in series.
•Inductive circuit: it is a circuit consisting of an
electric source and an inductor.
•Capacitive circuit: it is the circuit consisting of a
source of electrical energy and a capacitor C.

5. Simple Harmonic Motion
• Simple harmonic motion (SHM) refers to a
certain kind of oscillatory, or wave-like motion
that describes the behavior of many physical
phenomena:
– a pendulum
– a bob attached to a spring
– low amplitude waves in air (sound), water, the ground
– the electromagnetic field of laser light
– vibration of a plucked guitar string
– the electric current of most AC power supplies
_many more..

6. SHM Position, Velocity, and Acceleration

7. Simple Harmonic Motion
Periodic Motion: any motion of system
which repeats itself at regular, equal
intervals of time.

8. Simple Harmonic Motion

9. Simple Harmonic Motion
• Equilibrium: the position at which no net force acts on
the particle.
• Displacement: The distance of the particle from its
equilibrium position. Usually denoted as x(t) with x=0 as
the equilibrium position.
• Amplitude: the maximum value of the displacement with
out regard to sign. Denoted as xmax or A.

10. The period and frequency of a
wave
• the period T of a wave is the amount of time it takes to
go through 1 cycle
• the frequency f is the number of cycles per second
– the unit of a cycle-per-second is commonly referred to as
a hertz (Hz), after Heinrich Hertz (1847-1894), who
discovered radio waves.
• frequency and period are related as follows:
• Since a cycle is 2p radians, the relationship between
frequency and angular frequency is:
T
f
1

fp 2
T
t

11. Its position x as a function of time t is:
where A is the amplitude of motion : the distance from the
centre of motion to either extreme
T is the period of motion: the time for one complete cycle
of the motion.
Here is a ball moving back and forth with simple
harmonic motion (SHM):

12. NUMERICALS
1. A particle is executing SHM with amplitude 20 cm
and time 4 sec. find the time required by the
particle in passing between points which are at
distances 15cm and 5cm from center of force and
are on the same side of it. (t=0.38 sec)
2. A point executing S.H.M passing through two
points A and B 2m apart with the same velocity
having occupied 4sec is passing from A to B. after
another 4 sec it returns to B.find the period and
amplitude. (16sec, 1.141m)