Reference no: EM131045989
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.
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