1. Use separation of variables to find the analytical solution to the differential equation shown, with initial condition y(1) = 1.

y' = y^2/3/³√t

2. By hand and using a tabular format, use the Euler method to solve the differential equation through t = 8.5, with a step size of 2.5.

3. By hand and using a tabular format, use classical fourth order Runge-Kutta (RK4) to solve the differential equation through t = 8.5, with a step size of 2.5.

4. Create a table having 4 columns, with the leftmost column being t, and the 2^nd, 3^rd, and 4^th columns being the estimates of y at t, for t = 1,3.5, 6.0, and 8.5.

5. Plot on gridded paper the values of y from the Euler method, RK4, and the analytical solution, and sketch in, by eye, the best fit curve for each set of values. Which algorithm, Euler or RK4, provides better results. Briefly explain what it is about the algorithm that enables it to give better results.

6. The second order ODE shown is a model for the vibration of a building subjected to an earthquake. The earthquake p-waves (the back and forth movement of the ground in the horizontal direction) have frequency ω, and k and m represent the springlike stiffness and mass of the building, respectively.

y" + k/m.y = -ω²F_ocos(ωt)

a. Express the second order ODE as a system of two first order ODEs, in which y = y_A, and y'_A = y_B,

b. Assume ω = 2.5 radians/second, and that F_o, the amplitude of the p-wave, is 3 cm. Also, assume that the ratio k/m is 6.25/s², and that the horizontal displacement of the building and the horizontal velocity of the building at time t = 0 are both 0. Using an Excel spreadsheet, apply RK₄ to solve the system of ODEs through t = 30.0 seconds, using a time step of 0.25 seconds. If a horizontal displacement of 40 cm in either direction will cause the building to collapse, does this RK4 solution indicate that this building will collapse due to the earthquake? NOTE: To conserve paper, do not print out the entire RK4 table, but "HIDE" all but the first 5 rows and the row showing the peak displacement before printing out.

COMMENT: The RK4 solution when using the given step size very closely matches the analytical solution. But the Euler algorithm when using the given step size gives very poor results, showing a maximum displacement of approximately 8 miles!