Differential Equations - an introduction

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Differential Equations

A differential equation is an equation with derivatives in it, i.e. it relates derivatives to each other and/or to 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 such as this we need to resort to other methods than just a direct use of antiderivatives.

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

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Next page : A few examples from the kinematics

Last modified: Dec 28, 2023 @ 15:48