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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.

1. Euler's method is a numerical method for generating a table of values (x, , y) that approximate the 1. Euler's method is a numerical method for generating a table of values xytht approximate the solution of the differential equation y' = f(x,y) with boundary condition y(%)-yo. The first entry in the table is the starting point (x, , yo). Given the entry (x, , yi), then entry ( , yr) is obtained using the formula x1,-x; + Ax and yri-y, + xxf(x, , 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 Ax = 0.1, the exact solution of this differential equation is y-2x- x'. Compare your approximation values with the exact value