Note Recall if g a b R is an integrable vector valued functions with components 8 828 then 18 x dx 18 4 8 1 dx Equivalently written in the columnal 8 x dx a form we have jg x dx je x dx We will use these notations in this article Example 1 Obtain approximate solutions upto of the following initial value problem 2x 3y x 0 1 Thus Next dx dt dy dt Solution js 1 de 8 x dx 1 y y 0 2 We have x t 1 2 500 1 12 400 3 00 1 18 8 1 y 1 2 s y s ds 2 s 2 ds 2 21 2 21 1 2 2 81 1 1 2x s 3y s ds 1 8r 1 y 1 2 s y s ds 2 2 s Y s y 1 ds 2 2 3x 2 ds 13 21 2