Calculate the error between true solution-numerical solution

Question

Assignment:

Initial Value Problem 1:

Consider the following first order ODE:

dy/dt = t² - 3y/t from t =1 to t = 2.2 with y(1) = 1

(a) Solve with Euler's explicit method using .

(b) Solve with the midpoint method using .

(c) Solve with the classical fourth-order Runge-Kutta method using .

The analytical solution of the ODE is y = 1/6 (5/t³ +t³). In each part, calculate the error between the true solution and the numerical solution at the points where the numerical solution is determined.

Initial Value Problem 2:

Consider the differential equation for mass-spring-damper system as shown:

d²x/dt² + 2ydx/dt + k²x = 0

where k² = 48 N/m/kg, y = 0.7s^-1, x (0) = 0, and dx/dt|t=0 = 0.2m/s.

Solve the ODE over the ointerval  0≤ t ≤ 5 s, and plot x(t) amnd dx/dt (two separate figures on one page) as a function of t.

1.) Write the second order ODE as a system of first order ODEs

2.) Solve for step size t = 0.1 seconds

3.) Provide brief discussion of the physics (derivation of governing equation) and explanation of the results