1. Euler’s method is a numerical method for generating a table of values (xi , yi) that approximate the solution of the differential equation y’ = f(x,y) with boundary condition y(xo) = yo. The first entry in the table is the starting point (xo , yo.). Given the entry (xi , yi ), then entry (xi+1 , yi+1) is obtained using the formula xi+1 = xi + ?x and yi+1 = yi + ?x?f(xi , yi ). Where h is the small value called step size.

Use Euler’s method to estimate the value of y when x = 2.5 for the solution of the differential equation y’ = x + 3y/x with the boundary condition y(1) = 1. Take ?x = 0.1, the exact solution of this differential equation is y = 2x3– x2. Compare your approximation values with the exact value.

2. Twin primes are two primes that differ by 2 (e.g., 3 and 5, 101 and 103). Write a program that outputs all twin primes less than 1000 and the total count to a data file.