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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