### Differential equations: Introduction to Differential equations

### Notation for ODEs

Choosing an appropriate notation is often essential for performing mathematics. The theory of differential equations is no exception: we want a compact, readable notation for functions and derivatives.

Not only will we in this chapter mainly use *ordinary differential equations*, we will also most of the time limit ourselves to one unknown function for each equation. The unknown function is often denoted #y#. As mentioned above, the independent variable is often denoted #x# (for example, location, but also for other variables), or #t# (often for time).

Short notation for functions and derivatives

Because it is annoying having to write a variable \(y\) in big expressions as a function of \(t\), that is, by means of the function rule \(y(t)\), we use abbreviations: \(y(t)\) is abbreviated to \(y\) and \(y'(t)\) is written as \(y'\) or \(\displaystyle\frac{\dd y}{\dd t}\).

- The second derivative \(y''(t)\) is denoted \(y''\) or \(\displaystyle\frac{\dd^2y}{\dd t^2}\).
- If #n# is a natural number, then we write #y^{(n)}# both #y^{(n)}(t)# as #\frac{\dd^n}{\dd x^n} y(t)# the #n#-th derivative #y#.

Instead of the #n#-th derivative #y#, we also speak of the derivative of #y# of **order** #n#.

\[y'(t) = y'=y(t)'\]

Why would the use of #y(t)'# be less fortunate?

\[y'(2)=\left.\frac{\dd}{\dd t}(t^{4})\right|_{t=2}=\left.4 t\right|_{t=2}=8\]

and at the right-hand side

\[y(2)'=\left(2^{4}\right)'=\left({16}\right)'=0\]

which leads to #8=0#.

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