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Runge-Kutta 4

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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 - kv^, k>0.

Let: v(0) = 0, k = 0.125, m = 5 slugs, and g = 32 ft/s^2

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 +10sin(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)

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The solution shows in a step by step manner how to solve a first order differential equation using order 4 Runge-Kutta method and comparing it to the analytic solution

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Runge-Kutta Method Errors

How does this compare with the theoretical rate of convergence of 0(H2). Explain your result as best as you can?

We used the Runge-Kutta method to solve
y'(x) = - y(x) + x^0.1[1.1 + x], y(0) = 0
whose solution is y(x) = x^1.1. We solved the equation on [0,5] and we printed the errors at x = 1,2,3,4,5. We used stepsize h-0.1, 0.05, 0.0024, 0.0125, 0.00625. We calculated the ratios by which the errors decrease when h is halved.

The ratios by which the errors decrease when h is halved (see attached).

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