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
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
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
- Set up a system of differential equations for the amount of salt in
- Solve the system, and use the solution to find the concentration of
salt in each tank as the system approaches equilibrium (for large time).