Explain the steps on how to solve the problem below:
1. If air resistance is proportional to the square of the instantaneous velocity, then the velocity v of a mass m dropped from a given height is determined from
m(dv/dt)=mg− kv2, k >0
Let:
v(0)=0, k = 0.125,m = 5slugs and g = 32ft/s2
(a) Use a fourth-order Runge-Kutta (RK4) method with h = 1 to approximate the velocity v (5).
(b) Use Excel to graph the solution of the IVP on the interval [0, 6].
(c) Use separation of variables to solve the IVP and then find the actual value.
2. Find the analytic solution of the initial-value problem (IVP) y' = − y + 10 sin (3x), y(0) = 0, on the interval [0, 2].
(a) Graph the solution and find its positive roots.
(b) Use Runge-Kutta (RK4) method with h = 0.1 to approximate a solution of the initial-value problem (IVP) y' = − y + 10sin (3x), y (0) = 0.