Differential equations relate functions to their derivatives and define relationships between rates of change of physical quantities. They have applications in fields like medicine, engineering, chemistry, economics, and physics to model phenomena such as cancer growth, electricity movement, chemical reactions, optimal investment, and wave and pendulum motion. Specifically, population growth can be described by an exponential growth differential equation, and the leakage of water from a tin can punched with a hole is modeled by a differential equation where the leakage rate is directly proportional to the square root of the water depth.
2. Differential equations
A differential equation is a mathematical equation
that relates some function with its derivatives.
In applications, the functions usually represent
physical quantities, the derivatives represent their
rates of change, and the equation defines a
relationship between the two.
3. Applications of Differential equations
The applications of differential equations include:
In Medicine for modelling cancer growth or the spread of
disease
In Engineering for describing the movement of electricity
In Chemistry for modelling chemical reaction
In Economics to find optimum investment strategies
In Physics to describe the motion of waves, pendulums.
4. Differential equations in real time
Population Growth
Population growth is a dynamic process that can
be effectively described using differential equations.
The exponential growth of population can be
described by the differential equation,
=
where a is the growth rate
5. Tin can leakage problem
A tin can is filled with water then punch a
hole near the bottom. The water leaks
quickly at first, then more slowly as the
depth of the water decreases.
The rate at which the water leaks out its
directly proportional to the square root of its
depth.
=ky
