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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*=5*x*+*C*, where *C* is a constant. To check this we may differentiate the result to get y´=5 back.

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

In general, if we have

then the general solution is

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

These can be solved by taking the antiderivative *n* times.

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

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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