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A differential equation is an equation that relates one or more functions together, and the derivatives,

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many processes that occur in the real world can be expressed as differential equations.

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Now, these processes occur in nature.

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They occur in physics, biology, economics, engineering and many other areas.

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Isaac Newton invented calculus, and he used it to describe classical mechanics to express the relationship

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between precision, velocity and acceleration as a differential equation with respect to time.

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What this means, for example, is that if P represents the position of the object, then is velocity.

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V is how quickly the position is changing with time, i.e. the velocity V is the derivative of the position.

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P The same thing can be said about acceleration and velocity.

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Acceleration is the derivative of versity with respect to time, and it is the second derivative of

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position p with respect to time.

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And these specific equations are commonly referred to as the equations of motion.

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Differential equations can be broken down into two different types.

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There are ordinary differential equations or Odie's and there are partial differential equations.

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PDS, ordinary differential equations are an equation that contain only a single variable and is derivatives.

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Most problems encountered in physics are ordinary differential equations.

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For example, the equations of motion are all a function of derivatives of a single variable of precision.

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Heat is the independent variable and in the position, velocity and acceleration are dependent variables

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as they depend on time.

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These are the types of differential equations which we will focus on for data fusion and state estimation,

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as we usually want to track how variables evolve over time.

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With time being the independent variable, partial differential equations are equations that contain

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multiple variables and attributes.

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An example of this is the first equation which is used in many areas of physics, such as heat transfer

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and fluid dynamics.

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The equation order of a differential equation is determined by as highest derivative.

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If an equation only contains the first order derivatives, then it is a first order differential equation.

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Likewise, if it contains second order derivatives, then it is a second order differential equation.

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So going back to our equations of motion, the first equation for velocity is a first order differential

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equation, while the equation for acceleration is a second order differential equation, a linear system

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is a system that is output changes proportional to the input.

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And a nonlinear system is one that does not.

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Linear equations conform to the properties of an activity and homogeneity.

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A.R.T. just means that for a function, if you get the same result, if the inputs you and V added together

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before the function is evaluated, or if you evaluate the function separately with you and V and add

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the results together afterwards.

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Homogeneity just means that if the input is scaled, then it is the same as scaling the output of the

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function, a linear, ordinary differential equation means that both sides of the equation are linear,

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combination, order, dependent variables and derivatives.

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So most linear odors can be written in the following form.

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So here are two examples.

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One linear one and one non-linear one.

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The first one is linear as is dependent variable y and its derivatives are only linear combination of

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each other.

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While the second example is non-linear as the dependent variable Y is placed inside the nonlinear sine

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function.

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So the equation is no longer just a linear combination.