Math 212                                                  Problem Set #2                                    C.S.Davis


Please begin each problem on a separate sheet of paper. 

1. Give the solution of the linear homogeneous differential equation with constant coefficients (LHCC) whose auxiliary equation has the roots

                m = -2, 9,9,9, 4i, -4i, 0, 0,0, 6+3i, 6-3i, 6+3i, 6 -3i

2. Give the correct and most "economical" trial solution for the method of undetermined coefficients for this linear non-homogeneous differential equation. (Do not solve the DE; just give the trial solution.)

                y'' + 2y' - 15y = 12x e-5x + cos(x)

3. Solve this DE by the method of undetermined coefficients.

                2D2y - Dy - 6y = 56e2x

4. A. For the LHCC 3y'''' + 2y''' - 43y'' - 58y' +24y = 0,

        a. Give the possible rational roots for the auxiliary equation.

        b. Find the actual roots (m's) for the auxiliary equation.

        c. Give the general solution of the DE.

4. B. For the LHCC 7y'''' + 25y''' - 33y'' - 117y' + 54y = 0,

        a. Give the possible rational roots for the auxiliary equation.

        b. Find the actual roots (m's) for the auxiliary equation.

        c. Give the general solution of the DE.

5. Find the orthogonal trajectory of the family of curves below. Remember to eliminate the arbitrary constant.

                y8 = c e3x

6. Solve this DE by the Reduction of Order method.

                2D2y - Dy - 6y = 56e2x

7. Solve this DE by the Variation of Parameters method.

                2D2y - Dy - 6y = 56e2x

8. Enzymes in a certain detergent eat dirt and grow at a rate proportional to their mass at any time. Suppose that there are 4 grams of enzymes initially and after 7 minutes in the wash, the enzymes have grown to 10 grams. If there are 40 grams of dirt to eat in the wash, when will the enzymes finish the dirt and begin to eat the clothes? (Their mass will be 44 grams then.)

        a. Give the boundary value problem (BVP) describing the mass of the enzymes.

        b. Solve the BVP.

        c. Answer the question above.

9. A large tank contains 400 gallons of fruit punch in which 100 lbs. of sugar is dissolved. Syrup containing 3 lbs. of dissolved sugar per gallon runs into the tank at the rate of 7 gallons per minute. The mixture kept uniform by stirring runs out of the tank at 5 gallons per minute.

a. Give the IVP relating the amount of sugar, S, to time, t.

b. Solve the IVP. (You may use technology, write the answer and not show the work if you wish.)

        c. When will there be 450 lbs of sugar in the tank.

        d. On what time interval is this IVP valid.

10. In this problem, unsubstantiated correct guesses will be awarded minimal credit. Tell which theorem or definition is being cited.

a. Apply a theorem to determine whether it guarantees that the Cauchy-Euler IVP

        5x2y'' - 3xy' + 8y = 0,     y(1) = 3,     y'(1) = 6

has a unique solution. (Don't find the solution; just apply the appropriate theorem, and either  draw the conclusion or tell where the hypothesis is not satisfied.)
 

b. Apply a theorem or a definition to determine whether the functions

        f1(x) = x + 5,     f2(x) = x -2  and   f3(x) = 3x+1 

are linearly independent or linearly dependent.

11. Form an annihilator for the function

                f(x) = cos (3x) + 4 e5x - 6 x e4x + x2 e7x cos (8x).