We are going to look at linear, non-linear, differential, and quadratic
equations to why they are different and to see if there are any similarities.
First let's take a look at linear equations.
A linear equation is a form of algebraic equation that is
used to calculate the slope of a straight line. The results of the
calculations are then plotted on a Cartesian grid using the coordinates
derived from the calculation. The most common formula used in for
linear equations is y = mx + b. The point that is plotted on the Cartesian
grid is determined by the constant "m" that determines the slope and
the constant "b" will determine where the straight line crosses the
"y" axis on the grid.
Linear equations can also be used to make predictions on the outcome
of certain events. For example: you want to have a yard sale to get
rid of some of your old toys, video games, and other things. You will
look at your items and try to determine how much you will make. You
plot points on the graph based on your sales and then you can predict
how much you will make if sells continue the same.
Linear equations are used to find the slope on a graph. This is handy
when you add specific properties to the "x" and "y" axis. Now you
can pick any point on the line and find the outcome at the intersection
of the "x" and "y" axis. This is handy for making predictions and
is used in all forms of sports and business.
A non-linear equation is not a straight line and is essentially
difficult to solve. They are used primarily by mathematicians and
scientists. They use this type of equation to explain natural occurring
events, because very few things in life occur in a strait line. I
you have ever heard of the chaos theory, non-linear equations are
used to attempt to explain chaos.
Differential equations are used primarily in science and technology,
where relationships between events are continuously changing. For
example, Newton's Laws of Motion that explains the position, velocity,
acceleration, and forces that act on objects to make them move, stay
in motion, or rest. There are different types of differential equations:
- Ordinary - uses one single independent variable
- Partial - uses multiple independent variables
Differential equations are used to calculate how a leopard gets its
spots. Describing how diseases spread throughout a region or country.
In addition it is used by meteorologists to explain the patterns in
weather. Differential equations are also used in the development of
video games to predict and control specific actions within the game.
Quadratic equations 
are used to find the curve on a Cartesian grid. They are primarily
used to find the curve that objects take when the fly through the
air. For example a football, baseball, soccer ball, basketball, etc.
They also used to design any object that has curves that must have
a particular curved shape. The military uses the quadratic equation
when they want to predict where artillery shells hit when fired from
cannons. The quadratic equation is also used to explain how the planets
in our solar system revolve around the sun.
Do you see any comparisons between the different types of equations?
Not really, they are all used to calculate different things. Linear
equations are used for straight lines on a graph. None-linear are
used for determining the how non-straight things happen the way they
do. Differential equations are to explain changes in nature. Finally,
quadratic equations are used to explain curves.