MATH 215/255, Homework 7
d
1. Consider the statement: “ dt
u (t − a) = δ(t − a) ”. Give an argument why this is the case.
2. Solve
x00 = δ(t − 1),
x(0) = 1, x0 (0) = 0.
Sketch the graph of the resulting function.
3. Solve
x00 + x = δ(t − π) − δ(t − 2π),
x(0) = 0, x0 (0) = 1.
Sketch the graph of the resulting function.
4. Find the general solution to the system x0 = Ax, where A is as specified below. Make sure to write
the solution in purely real form.
1 0
(a) A =
.
−4 3
1 −5
(b) A =
.
1 −3


1
0 0
(c) A =  −2 1 2 
−2 −2 1
−5 −3
(d) A =
3
1
5. A 3x3 real
has three eigenvalues. One of them is λ1 = −1 and its corresponding
eigenvector


 matrix
1
0
is v1 =  1  . Another is λ2 = 1 + i and its corresponding eigenvector is v2 =  2  . (a) What
i
0
is the third eigenvalue and its corresponding eigenvector? (b) Find the
general
solution
to x0 = Ax.
 
1
(c) Find the solution to x0 = Ax subject to initial condition x (0) =  0  .
0
6. Two tanks are connected by two pumps: one pump pushes the liquid from tank A to tank B at
the rate of 6 liters/hour while the other pushes from tank B to tank A at the same rate. Initially,
both tanks contain 2 liters of liquid and tank A contains pure water while tank B has a mixture of
80% water and 20% pollutant. (a) Find the concentration of pollutant in tanks A and B after ten
minutes. (b) Find the concentration of pollutant in tanks A and B after a very long time.
1