Two tanks each contain 100 liters of a mixture. Initially, the mixture in tank A

Find the general solution of each of the following systems: (a) y_ 1 = y1 + y2 y_ 2 = 2y1 + 4y2 (b) y_ 1 = 2y1 + 4y2 y_ 2 = y1 3y2 (c) y_ 1 = y1 2y2 y_ 2 = 2y1 + 4y2 (d) y_ 1 = y1 y2 y_ 2 = y1 + y2 (e) y_ 1 = 3y1 2y2 y_ 2 = 2y1 + 3y2 (f) y_ 1 = y1 + y3 y_ 2 = 2y2 + 6y3 y_ 3 = y2 + 3y3

Solve each of the following initial value problems: (a) y_ 1 = y1 + 2y2 y_ 2 = 2y1 y2 y1(0) = 3, y2(0) = 1 (b) y_ 1 = y1 2y2 y_ 2 = 2y1 + y2 y1(0) = 1, y2(0) = 2 (c) y_ 1 = 2y1 6y3 y_ 2 = y1 3y3 y_ 3 = y2 2y3 y1(0) = y2(0) = y3(0) = 2 (d) y_ 1 = y1 + 2y3 y_ 2 = y2 y3 y_ 3 = y1 + y2 + y3 y1(0) = y2(0) = 1, y3(0) = 4

Given Y = c1e1tx1 + c2e2tx2 + +cnen txn is the solution to the initial value problem: Y_ = AY, Y(0) = Y0 (a) Show that Y0 = c1x1 + c2x2 + +cnxn (b) Let X = (x1, . . . , xn) and c = (c1, . . . , cn)T . Assuming that the vectors x1, . . . , xn are linearly independent, show that c = X1Y0.

Two tanks each contain 100 liters of a mixture. Initially, the mixture in tank A contains 40 grams of salt while tank B contains 20 grams of salt. Liquid is pumped in and out of the tanks as shown in the accompanying figure. Determine the amount of salt in each tank at time t. Mixture 16 L/min Mixture 12 L/min Mixture 4 L/min Water 12 L/min B 100 L A 100 L

Find the general solution of each of the following systems: (a) y__ 1 = 2y2 y__ 2 = y1 + 3y2 (b) y__ 1 = 2y1 + y_ 2 y__ 2 = 2y2 + y_ 1

Solve the initial value problem y__ 1 = 2y2 + y_ 1 + 2y_ 2 y__ 2 = 2y1 + 2y_ 1 y_ 2 y1(0) = 1, y2(0) = 0, y_ 1(0) = 3, y_ 2(0) = 2

In Application 2, assume that the solutions are of the form x1 = a1 sin t, x2 = a2 sin t. Substitute these expressions into the system and solve for the frequency and the amplitudes a1 and a2.

Solve the the problem in Application 2, using the initial conditions x1(0) = x2(0) = 1, x_ 1(0) = 4, and x_ 2(0) = 2

Two masses are connected by springs as shown in the accompanying diagram. Both springs have the same spring constant, and the end of the first spring is fixed. If x1 and x2 represent the displacements from the equilibrium position, derive a system of second-order differential equations that describes the motion of the system. m1 m2 x1 x2 1

Three masses are connected by a series of springs between two fixed points as shown in the accompanying figure. Assume that the springs all have the same spring constant, and let x1(t), x2(t), and x3(t) represent the displacements of the respective masses at time t. A B m1 m2 m3 (a) Derive a system of second-order differential equations that describes the motion of this system. (b) Solve the system if m1 = m3 = 1 3 , m2 = 1 4 , k = 1, and x1(0) = x2(0) = x3(0) = 1 x_ 1(0) = x_ 2(0) = x_ 3(0) = 0 1

Transform the nth-order equation y(n) = a0 y + a1 y_ + +an1 y(n1) into a system of first-order equations by setting y1 = y and yj = y_ j1 for j = 2, . . . , n. Determine the characteristic polynomial of the coefficient matrix of this system.