Unlimited growth: Linear and quadratic growth
Cubic growth and beyond
Properties of cubic growth For cubic growth of a quantity \(y(t)\) at time \(t\) :
- the second derivative of quantity \(y(t)\) changes with the same value per unit of time ;
- \(y(t)\) can be described by a cubic function \(y(t)=a\cdot t^3+b\cdot t^2 + c\cdot t +d\), where \(6a\) is the increase or decrease of the second derivative \(y''(t)\) per unit of time, \(2b\) the value of the second derivative at time \(t=0\) ( \(y''(0)=2b\) ), \(c\) the value of the first derivative at time \(t=0\) ( \(y'(0)=c\) ) and \(d\) the initial value at time \(t=0\);
- the corresponding graph of \((y,t)\) is a third-degree polynomial function;
- the third derivative of the quantity \((y,t)\) is a constant.
So cubic growth of a quantity \(y(t)\) is described by: \(y'''(t)=6a\), for some constant \(a\). This is a differential equation, because it has a derivative. In this case, the third derivative plays a role; in mathematical jargon, this is a differential equation of order three.
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Of course, we can continue like this for a while: linear growth, quadratic growth, cubic growth, quartic growth, ... A further generalisation of the dynamic model of linear growth is the differential equation \[y'(t)=f(t),\] where \(f\) is a function. If a function \(F\) is known such that the derivative of \(F\) equals \(f\) (so \(F\) is the antiderivative of \(f\)), then the general solution of the differential equation is given by \[y(t)=F(t)+c\] for some constant \(c\). So the solution is obtained by finding antiderivative of \(f\), i.e., a function \(F\) with the property that \(F'(t)=f(t)\) for all \(t\). Finding antiderivatives is also called integrating functions. Integrating functions is often required for solving other types of differential equations. The graph of a solution of a differential equation is therefore often referred to as an integral curve.
