Notation for ODEs

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#.

Thus, the function rule #y(t)# and the function #y# are confused on purpose. The context often makes clear exactly what is meant.

All of the following ways of writing refer to the same ODE:

#\frac{\dd^2}{\dd t^2} y(t)+5t\cdot \frac{\dd}{\dd t} y(t) +y(t) =3 #
#\frac{\dd^2}{\dd t^2} y+5t\cdot \frac{\dd}{\dd t} y +y =3 #
# y''(t)+5t\cdot y'(t) +y(t) =3 #
# y''+5t\cdot y' +y=3 #

The statement \[y\text{ is a function of }t\] is sometimes written for short as \[y=y(t)\] This is the same abbreviation as we introduced the theory. If we also use this abbreviation for #y'#, we get
\[y'(t) = y'=y(t)'\]
Why would the use of #y(t)'# be less fortunate?

Take #y(t)=t^{3}#. If replace #t# by #2# and look at what happens to the statement #y'(2)=y(2)'#, then we find at the left-hand side
\[y'(2)=\left.\frac{\dd}{\dd t}(t^{3})\right|_{t=2}=\left.3 t\right|_{t=2}=6\]
and at the right-hand side
\[y(2)'=\left(2^{3}\right)'=\left({8}\right)'=0\]
which leads to #6=0#.

Therefore, we try to avoid writing #y(t)'# for the derivative of #y# with respect to #t#.

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