(a) Describe the Newton-Raphson method for finding the cube root of a…

Question

(a) Describe the Newton-Raphson method for finding the cube root of a real number $N$. Apply it to calculate the cube root of $18$, correct to three decimal places.

(b) Using Picard's method, find the approximate value of $y$ at $x = 0.2$, given that $y' = x + y + xy$ with $y(0) = 1$.

(c) Using Euler's method, evaluate $y(0.8)$, given that $y' = x^2 + y^2$ with $y(0) = 1$. Take $h = 0.2$.