Mathematical Analysis: Linear Differential Equations

Abstract

There are many ways in which differential equations arise in practice but they all involve continuous variation. Among the simpler forms of differential equations are those representing a cumulative growth when the addition is made continuously and in a specified way. Continuity here is a concept obtained from finite processes by proceeding to the limit, just as the derivative of a function is defined from finite changes. This is illustrated by considering a sum of money accumulating at compound interest at the rate of 100r% per year. Let £Y ₀ be the initial sum, amounting to £Y _x at the end of x years. If interest is compounded annually, then Y _x+1 = (1 + r)Y _x where x takes the values, 0, 1, 2, 3, … ; if compounding is twice a year, then ${Y_{x + \frac{1}{2}}} = \left( {1 + \frac{1}{2}r} \right){Y_x}{\text{ }}where{\text{ }}x = 0,\frac{1}{2},1,\frac{3}{2}, \ldots .$ Generally, if interest is compounded n times a year and x runs through the values $0,\frac{1}{n},\frac{2}{n},\frac{3}{n}, \ldots ,$, then :

$${Y_{x + \frac{1}{n}}} = \left( {1 + \frac{r}{n}} \right){Y_x}$$