A differential equation is an equation that relates one or more functions together, and the derivatives,
many processes that occur in the real world can be expressed as differential equations.
Now, these processes occur in nature.
They occur in physics, biology, economics, engineering and many other areas.
Isaac Newton invented calculus, and he used it to describe classical mechanics to express the relationship
between precision, velocity and acceleration as a differential equation with respect to time.
What this means, for example, is that if P represents the position of the object, then is velocity.
V is how quickly the position is changing with time, i.e. the velocity V is the derivative of the position.
P The same thing can be said about acceleration and velocity.
Acceleration is the derivative of versity with respect to time, and it is the second derivative of
position p with respect to time.
And these specific equations are commonly referred to as the equations of motion.
Differential equations can be broken down into two different types.
There are ordinary differential equations or Odie's and there are partial differential equations.
PDS, ordinary differential equations are an equation that contain only a single variable and is derivatives.
Most problems encountered in physics are ordinary differential equations.
For example, the equations of motion are all a function of derivatives of a single variable of precision.
Heat is the independent variable and in the position, velocity and acceleration are dependent variables
as they depend on time.
These are the types of differential equations which we will focus on for data fusion and state estimation,
as we usually want to track how variables evolve over time.
With time being the independent variable, partial differential equations are equations that contain
multiple variables and attributes.
An example of this is the first equation which is used in many areas of physics, such as heat transfer
and fluid dynamics.
The equation order of a differential equation is determined by as highest derivative.
If an equation only contains the first order derivatives, then it is a first order differential equation.
Likewise, if it contains second order derivatives, then it is a second order differential equation.
So going back to our equations of motion, the first equation for velocity is a first order differential
equation, while the equation for acceleration is a second order differential equation, a linear system
is a system that is output changes proportional to the input.
And a nonlinear system is one that does not.
Linear equations conform to the properties of an activity and homogeneity.
A.R.T. just means that for a function, if you get the same result, if the inputs you and V added together
before the function is evaluated, or if you evaluate the function separately with you and V and add
the results together afterwards.
Homogeneity just means that if the input is scaled, then it is the same as scaling the output of the
function, a linear, ordinary differential equation means that both sides of the equation are linear,
combination, order, dependent variables and derivatives.
So most linear odors can be written in the following form.
So here are two examples.
One linear one and one non-linear one.
The first one is linear as is dependent variable y and its derivatives are only linear combination of
each other.
While the second example is non-linear as the dependent variable Y is placed inside the nonlinear sine
function.
So the equation is no longer just a linear combination.