First Order Differential Equations

¶In many fields such as physics, biology or business, a relationship is often known or assumed between some unknown quantity and its rate of change, which does not involve any higher derivatives. It is therefore of interest to study first order differential equations in particular.Definition 5.7. First Order DE.A first order differential equation is an equation of the form \(F(t, y, y')=0\text{.}\) A solution of a first order differential equation is a function \(f(t)\) that makes \(\ds F(t,f(t),f'(t))=0\) for every value of \(t\text{.}\) Here, \(F\) is a function of three variables which we label \(t\text{,}\) \(y\text{,}\) and \(y'\text{.}\) It is understood that \(y'\) will explicitly appear in the equation although \(t\) and \(y\) need not. The variable \(y\) itself is dependent on \(t\text{,}\) hence it is understood that \(y'\) must be the derivative of \(y\) with respect to \(t\text{.}\) Since only the first derivative of \(y\) appears, but no higher order derivative, this is a...