Assignment-

Initial Value Problem 1:

Consider the following first order ODE: 

dy/dt = t² - 3y/t                from t = 1 to t = 2.0 with y(1) = 1

(a) Solve with Euler's explicit method using h = 0.4.

(b) Solve with the midpoint method using h = 0.4.

(c) Solve with the classical fourth-order Runge-Kutta method using h = 0.4.

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 + 2γ(dx/dt) + k²x = 0      

where k² = 48 N/m/kg, γ = 0.7s^-1, x(0) = 0, and dx/dt|_t=0 = 0.2 m/s. Solve the ODE over the interval 0 ≤ t ≤ 5 s, and plot x(t) and dx/dt 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.