# Review Topics for Exam 2

• Characteristic polynomials, eigenvalues, eigenvectors
Page 110 Number 7, 9, 20

• Homogeneous systems, general solutions, initial value problems
Page 126 Number 1, 4, 9

• Matrix solutions, fundamental matrix
Page 135 Number 1, 16, 17, 24

• Inhomogeneous systems, undetermined coefficients, variation of parameters
Page 149 Number 6, 7, 9, 10
Page 154 Number 1, 2

• Modeling with systems
Two 100 gallon tanks are initially filled with pure water. Fluid is exchanged between the two tanks through a pair of pipes, through which fluid is pumped in opposite directions at a rate of 5 gallons per minute. At the same time, a brine solution containing 5 pounds of salt per gallon is pumped into tank 1 at a rate of 10 gallons per minute, and pure water is pumped into tank 2 at a rate of 5 gallons per minute. The overflow from each tank is allowed to drain off, so that the volume of fluid in each tank remains constant.
1.
Set up a system of differential equations for the amount of salt in each tank.
2.
Solve the system, and use the solution to find the concentration of salt in each tank as the system approaches equilibrium (for large time).

Frank Beatrous
1998-03-09