Consider the following initial value problems, the first three on the interval [0, 1], and the fourth one on the interval [0, 40]:

y'+ y = 0 y(0) = 1

y' y^2 = t^2 y(0) = 0

( y'= 4y + 3z z' = 2y z y(0) = 1, z(0) = 1

y'' + (y^2 1)y'+ y = 0 y(0) = 0.1, y'(0) = 0.1

a) How many discretization points do you need to find the solution to 3 exact digits (absolute error less than 510^4 ) when solving the differential equations using the forward Euler method? Produce a table with 4 rows (corresponding to the differential equations) containing the corresponding number of discretization points required (exact up to a factor of two, e.g., if 150 points are required, then 250 is considered a correct answer, but 301 is not).

b) Plot your solutions exact to 3 digits separately (one figure for each differential equation).