Math 212 Problem Set #5

Math 212 Problem Set #5 C. S. Davis

Please begin each problem on a separate sheet of paper. (For uniformity, round all intermediate and final numerical values to the fourth decimal place.)

1. Use the Euler method for n = 2 iterations (h = 0.5) to solve the IVP,

y' = 3x - y, y(1) = 4, numerically on [1,2].

2. Use the Improved Euler method for n = 2 iterations (h = 0.5) to solve the IVP,

y' = 3x - y, y(1) = 4, numerically on [1,2].

3. Use the second order Taylor method for n = 2 iterations (h = 0.5) to solve the IVP,

y' = 3x - y, y(1) = 4, numerically on [1,2].

4. Use the fourth order Runge-Kutta method for n = 2 iteration (h = 0.5) to solve the IVP,

y' = 3x - y, y(1) = 4, numerically on [1,2].

5. a. Solve the IVP above algebraically. It's a first order linear.

b. Find y(2) so that you can see how close the graphical and various numerical methods came to the actual value of y(2).

c. Produce a small table indicating the four methods used and their results. Write a paragraph declaring the relative merits for the five methods and some reason(s) for using more than one method.

6. Solve this system of linear IVP's numerically on the interval [0,2], using the Euler method with n = 2 iterations (h = 1).

x' = 2x - y - t, x(0) = 2
y' = 3y + t, y(0) = -1

7. Solve this system of linear IVP's numerically on the interval [0,2], using the Runge-Kutta method with n = 1 iteration (h = 2).

x' = 2x - y - t, x(0) = 2
y' = 3y + t, y(0) = -1

8. Give the Fourier series for the periodic function

                       { 2        for -4 <= x < 0 }
             f(x) = {                                    } ,           f(x + 8) = f(x)
                       { x+2    for 0 <= x < 4  }

9. a. Find the general product solution to the partial differential equation by the method of separation of variables.

U_x = x y² U_y - 2U

b. Apply the boundary condition U(0,y) = 4 e^6/y to the general product solution above to find the solution
to the Boundary Value Problem.