Math 212                                  Problem Set #3                                      C.S.Davis

 

Please begin each problem on a new sheet of paper.      Name_______________________

(Turn in at least 7 sheets of paper)

1. Obtain the general solution this system of linear differential equations by the operator method. Eliminate y to solve for x first. Recall that differentiation is performed with respect to  t.

    a. Write the system in differential operator notation in "good form" .

    b. Eliminate y and solve the system for xc, xp and xg.

    c. Find yg.

    d. Check xg and yg in both DE=s of the original system.

            x' = 3x + y + 9

            y' = 12x + 2y

2. Convert this initial value problem to a system of first order IVP's in normal form. Draw a box around your complete answer: the DE=s and the IC=s. Use the variables, z1, z2, z3 and z4, as your new dependent variables.

            y'''' - 9t y''' -8sin(t) y'' + 7t3 y' - 2 y  =  3 e5t

            y(0) = 3,     y'(0) = 4,    y''(0) = 5,     y'''(0) = 6

3. A 96 pound weight stretches a large spring 6.4 feet. A resisting medium exerts a resisting force of 6 times the velocity. The weight is set in motion from its equilibrium point with an initial velocity of 12 feet per second downward.

    a. Give the initial value problem (DE and IC's) describing the motion.

    b. Solve the differential equation to get the general solution.

    c. Apply the initial conditions to solve the IVP.

    d. Name and describe the motion and sketch a rough graph of the solution.

4. a. Solve this Cauchy-Euler DE with variable coefficients.

            3x2y'' - 10xy' + 4y = 0

    b. Check the general solution by substituting it in the original DE.

    c. Suppose the roots of the auxiliary equation for another Cauchy-Euler DE are

        m = 9,9,9,9. Give the general solution to the DE (for the x>0 case).

    d. Suppose the roots of the auxiliary equation for another Cauchy-Euler DE are

        m = 8+5i and 8 - 5i. Give the general solution to the DE (again for the x>0 case).

5. Consider the LRC circuit below. Assume that when the switch is closed, the charge on the capacitor is 1 coulomb and the current is 0 amps.

    a. Give the initial value problem (DE and IC's) describing the circuit.

    b. Use Maple to solve the IVP and graph the solution.

    c. Experiment with lots of different window sizes (plot ranges) to determine a graph that the  steady state of the circuit achieves. (SepVar.mws is a good reference for the format of Maple=s dsolve and plot functions. 

In one loop, there is an EMF (voltage source) of  170 sin(120 Pi t),  an inductor of 3 henrys,  a resistor of 6 ohms, a capacitor of 0.02 farads and a switch that closes at time zero.

 

6. Give a series solution for the IVP

            (1 - x2)y'' - 2xy' + 56y = 0,      y(0) = 0,     y'(0) = 1

(This is Legendre's equation for p = 7, a fact that you do not need to know to solve the IVP.)

7. a. Find the series general solution for the following DE expanded about x0 = 0:

            y' - 3x2y = 0

    b. Solve the DE by separation of variables. Then use the series for ex in the solution appropriately to show that the two methods verify each other.