Choosing an appropriate notation is often essential for performing mathematics. The theory of differential equations is no exception: you want a compact, readable 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 functional expression \(y(t)\), 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 written as \(y''\) or \(\displaystyle\frac{\dd^2y}{\dd t^2}\). If \(n\) is a natural number, then we denote the \(n\)-th derivative of \(y(t)\) as \(y^{(n)}\) or \(\displaystyle\frac{\dd^ny}{\dd t^n}\).
In this short notation, the ODE of exponential growth is \[\frac{\dd y}{\dd t}=r\cdot y\] This short notation \(\displaystyle\frac{\dd y}{\dd t}\) for a derivative matches with the concept of differential: the latter ODE can be written in terms of differentials as \[\dd y=r\cdot y\,\dd t\] We call this "writing down the differential equation in differential form." How useful this notation will become evident when we determine the solution of this ODE using the method of separation of variables.
Suppose the function \(y=f(t)\) is differentiable in the point \(t\), that is, has a non-vertical tangent line in this point. For a small increase \({\vartriangle}t\) of \(t\), one can estimate the increase \({\vartriangle}y=f(t+{\vartriangle}t)-f(t)\) well by the following formula: \[{\vartriangle}y\approx f'(t)\cdot {\vartriangle}t\] The closer in the neighbourhood of \(t\), that is, the smaller \({\vartriangle}t\), the better is the estimated value of the function. The underlying idea of this formula is that the graph of a smooth function is in practice almost a straight line when one sufficiently zoomes in. For negligible changes, denoted with \(\dd y\) and \(\dd t\) and called infinitisimal changes, the following statement can be written down:
\[\text{If }y=f(t), \text{ then }\dd y=f'(t)\,\dd t\] The right-hand side \(f'(t)\,\dd t\) is called the differential of f and one uses for this the notations \(\dd y\), \(\dd f\) and \(\dd\bigl(f(t)\bigr)\).
The differential \(f\) therefore depends on the time \(t\) and the infinitesimal change \(\dd t\). If \(\dd y\) is denoted as \(df\), the relationship between differentials and derivatives is easily made: the quotient of the differentials \(df\) and \(\dd t\), also known as differential quotient is equal to the derivative \(f'(t)\) in \(t\). Hence the mixed use of \(y'\) and \(\displaystyle\frac{\dd y}{\dd t}\) for the derivative function in this instructional material.
It is thus clear that a differential equation, that is, an equation in which besides a yet unknown function also one or more derivatives of that function are present can also be written as an equation between differentials: for example, the differential equation \(\displaystyle\frac{\dd y}{\dd t}=y\) can be written as a relationship between differentials as \(\dd y=y\,\dd t\). More generally, the differential equation \(\displaystyle\frac{\dd y}{\dd t}=\varphi(t,y)\) with \(\varphi\) a function of two variables, can be rewritten in terms of differentials as \(\dd y=\varphi(t,y)\,\dd t\).
In short, the method of separation of variables boils down in the example of exponential growth with ODE \[d y=r\cdot y\,\dd t\] to rewriting the differential form by ― nomen est omen ― separating the variables \(y\) and \(t\) as \[\frac{\dd y}{y}= r\,\dd t\] Integrating the left- and right-hand sides of the equation \[\int \frac{1}{y}\,\dd y= \int r\,\dd t\] leads to \[\ln|y|=r\cdot t +C\] Rewriting of this equation leads to: \[|y|=e^{r\cdot t+C}=e^C\cdot e^{r\cdot t}\] The general solution of the ODE \[y'=r\cdot y\] is then \[y=c\cdot e^{r\cdot t}\] where \(c\) is a constant.
Notation for ODEs
