# The sum of Rs $1000$ is compounded continuously, the nominal rate of interest being four percent per annum. In how many years will the amount be twice the original principal ? $(\log_{e}$$2=0.6931) ## 1 Answer Toolbox: • First order,first degree DE variable separatable : Variable of a DE are rearranged to separate there (i.e.) f_1(x)g_2(y)dx+f_2(x)g_1(y)dy=0 • It can be written as \large\frac{g_1(y)}{g_2(y)}$$dy=\large\frac{f_1(x)}{f_2(x)}$$dx • The solution is therefore \int \large\frac{g_1(y)}{g_2(y)}$$dy=-\int \large\frac{f_1(x)}{f_2(x)}$$dx+c Step 1: The sum of Rs1000 is the principal or initial amount.(x_0) Step 2: Let x be the amount at time t years,the amount being compounded continuously,at an annual rate of 4% per annum. Step 3: \therefore rate of change of x is 0.04 \large\frac{dx}{dt}$$=0.04x$
$\large\frac{dx}{x}$$=0.04dt \int \large\frac{dx}{x}$$=\int 0.04 dt+\log c$
$\log x=0.04t+\log c$