Differential Equations - an introduction

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Differential Equations

A differential equation is an equations with derivatives in them, i.e. it relates derivatives to each other and/or to a functions and variables. To solve a differential equation is to find a function or functions that satisfies the equation. The simplest example might be something like y´=5. Here we could simply take the antiderivative of both sides to get y=5x+C, where C is a constant. To check this we may differentiate the result to get y´=5 back.

The solution y=5x+C is called the general solution and say y=5x+2 is a so called particular solution.

In general, if we have

$y' = f(x)$

then the general solution is

$y = F(x) + C = \int {f(x)dx}$

The highest order of derivatives in the equation is the order of the equation. The simplest differential equation of order n is of the form

${y^{(n)}} = f(x)$

These can be solved by taking the antiderivative n times.

$y = \underbrace {\int {...\int \; } }_nf(x)d{x^n}$

Differential equations may be a lot more complicated though. We could have a mix of orders, and powers like in

$y'' + 3y' + y = x$

To solve equations as this we need to resort to other methods than just a direct use of antiderivatives.

Differential equations is an absolutely necessary part of the fundament of modern physics, and it is extensively used in engineering, modelling of biological phenomena, in advanced economic analysis and so on. I would go so far to say that differential equations is a corner stone of our civilisation. One of the most famous differential equation might be the Schrödinger equations that is one of the fundamental equations of the quantum physics.

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Next page : A few examples from the kinematics

Last modified: Mar 13, 2019 @ 20:10