Differential Equations

*If you haven’t, please check this note here on the Science page.

Differential equations are the type of equations that describe dynamic systems, whether they are naturally existing in the vast cosmos or humanly engineered in our universe. Such  systems are characterized by having many variables that are changing with respect to others and therefore, the algebraic equations of school math are not the appropriate tool to use and the need for describing such variations arise. To describe variations, we recall using calculus for this, or more specifically differentiation, which is simply defined as the rate of change of a certain variable with respect to another; a simple example may be the rate of change of temperature with time in the event of heating a steel bar for instance. Of course, in large systems there are various variables and several changes (or derivatives) and the set of equations become exceptionally big one and in many cases, simplification is required to treat the system mathematically.

We’ll start our discussion here by classifying the differential equation and describing the meaning of a solution to the equation. Then begin working with first order differential equations and then move on up to partial differential equations.

 

Basic Concepts and Ideas

So, let’s start with some basic concepts and ideas. Perhaps the first classification you would encounter in the world of differential equations is the ordinary-vs-partial differential equations. “Ordinary Differential Equation” (ODE) is an equation that contains 1 or several derivatives of unknown function, which we call y(x) and actually solving the ODE is simply determining this function. However, the main characteristic of an ODE is that the derivatives here are representing a change of a variable with respect to “one” variable, for instance if y is temperature, then we know for sure that y changes only with time or location for example, but it does NOT change with both.
Now it’s time to see how an ODE looks like. These are some examples of simple ODEs: y'=cos x  ,  y''+4y=0  and  x^2 y'' y' + 2 e^x y'' = y^2

At the other side, a “partial” differential equation (PDE) contains derivatives that imply the change (derivative) with respect to multiple variables. If we consider the temperature variable example again, then here temperature may be changing with time, space, pressure, … To get a feeling of the ongoing discussion, here are some examples for PDEs: \frac{\partial z}{\partial y}+\frac{\partial z}{\partial x} = \frac{\partial u}{\partial t} and  \frac{\partial v}{\partial t}+c^2\frac{\partial \rho}{\partial x} = \frac{\partial P}{\partial y} + u^4

Next important classification is the “Order” of a DE. This is the order of the highest derivative appearing in the equation. “First-order DE” is the one that contains only the first derivative (y’ for example with respect to y) and may contain also (y) & f(x). Second order DE is the one with second derivative (y” for instance). Higher order DEs follow same sequence.

 

1) Ordinary Differential Equations

2) Partial Differential Equations

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