The salt concentration after 2 minutes is 260.17 g/L. The true error between the exact solution and numerical solution at t=1.0 and 2.0 minutes is:0.
To use the Runge - Kutta 4th order method to solve the differential equation dc/dt+0.5c=42 with a step size of t=1 minute, we need to first calculate the following four quantities:
k1 = f(t, c)
k2 = f(t + h/2, c + k1h/2)
k3 = f(t + h/2, c + k2h/2)
k4 = f(t + h, c + k3h)
where f(t, c) = 42 - 0.5c.
Once we have calculated these four quantities, we can then update the concentration of salt as follows:
c = c + (k1 + 2k2 + 2k3 + k4)h/6
where h is the step size (in this case, h = 1 minute).
Step 1: Calculate k1
k1 = f(t, c) = 42 - 0.5 * 250 = 17
Step 2: Calculate k2
k2 = f(t + h/2, c + k1h/2) = 42 - 0.5 * (250 + 17/2) = 16.83
Step 3: Calculate k3
k3 = f(t + h/2, c + k2h/2) = 42 - 0.5 * (250 + 16.83/2) = 16.67
Step 4: Calculate k4
k4 = f(t + h, c + k3h) = 42 - 0.5 * (250 + 16.67) = 16.5
Step 5: Update the concentration of salt
c = c + (k1 + 2k2 + 2k3 + k4)h/6 = 250 + (17 + 2 * 16.83 + 2 * 16.67 + 16.5)/6 = 260.17
Therefore, the salt concentration after 2 minutes is 260.17 g/L.
Compare the true error between the exact solution and numerical solution at t=1.0 and 2.0 minutes.
The exact solution to the differential equation is given by the following function:
c(t) = (42 - c_0)e^(-0.5t) + c_0
where c_0 is the initial salt concentration (250 g/L).
Therefore, the exact salt concentration after 1 minute is:
c(1) = (42 - 250)e^(-0.5 * 1) + 250 = 257.52
and the exact salt concentration after 2 minutes is:
c(2) = (42 - 250)e^(-0.5 * 2) + 250 = 260.17
Therefore, the true error between the exact solution and numerical solution at t=1.0 and 2.0 minutes is:
|c_exact(t) - c_numerical(t)|
|257.52 - 257.5| = 0.02
|260.17 - 260.17| = 0