Name_____________________________________
Math 44 – Differential Equations
Friday, March 23, 2007
Graded Problem Set 2 - SOLUTIONS
Due at start of class Friday, March 2423, 2007. This problem set substitutes in some measure
for an exam. You may not consult with anyone other than the instructor, but you may use
references (including the text and your notes), calculators or computers, and any
software. (There are 9 numbered problems on 3 pages.)
All about y'' + by' + cy = 0
1. Find two linearly independent solutions of
y '' 6 y ' y  0 .
The characteristic equation is 2+6+1, and its roots are 3  8 and
3  8 . These are both real numbers. So two linearly independent
solutions are
 3 8 t
e
and
 3 8 t .
e
2. Find two linearly independent solutions of
numbers in the solution itself.
y '' 6 y ' 19 y  0 .
Don’t use complex
Now the roots are 3  i 10 and 3  i 10 , a complex pair. So two
solutions are
e3t cos( 10 t )
and
e3t sin( 10 t ) .
3. About repeated eigenvalues: Suppose that the characteristic polynomial of
y '' 2 y ' cy  0 has exactly one eigenvalue, with multiplicity two.
a. What is c ?
The discriminant of the polynomial is (–2)2 – 4c, which must be zero for
there to be a repeated root. Solving, c = 2.
b. What is the eigenvalue ?
It’s .
c. Find one eigenvector, V1.
0  
Solve 

2
 
  k1   0 
 k1   1 
      to get     
2     l1   0 
 l1    
1
(or any non-zero multiple of that).
1
d. Either: (1) find another eigenvector V2, linearly independent of V1, or
(2) find a “generalized eigenvector” V1’ satisfying the “generalized
eigenvalue equation” (A-I)V2=V1. (Here A is the matrix of the
1 
0
 .)
 c 2 
system related to the equation; that is, A= 
There aren’t any other eigenvectors, so we need to solve
0  
 2
 
  k2    
 k2   0 
      to get, for example,      .
2     l2   1 
 l2    
1
We don’t need the generalized eigenvector to solve the equation; we know
the solutions by rote as soon as we have the repeated eigenvalue . This
problem is just to make the point that when repeated eigenvalues occur in a
system that comes from a higher-order linear equation, we’re always in the
only-one-eigenvector case.
e. Pick one: When a system arising from the equation y '' by ' cy  0
has a repeated eigenvalue, there are
___ always two linearly independent eigenvectors
___ sometimes two linearly independent eigenvectors
X never two linearly independent eigenvectors.
f. Find two linearly independent solutions for
y '' 2 y '  2 y  0 .
e t and tet .
Practice with the identity cos ( x + y ) = cos x cos y – sin x sin y
4. Let  be a fixed, non-zero constant. Show that the family of functions
f(t) = c1 cos (t) + c2 sin (t)
is the same as the family of functions
g(t) = A cos ( ( t -  ) ).
That is, show that any function that can be written in the first form for some c1, c2 can
also be written in the second form for some A and , and vice versa.
2
Start with g(t), and use the facts that cos(–)=cos(), sin(–)=–sin():
g(t) = A cos ( t –  ) = A ( cos(t) cos() + sin(t) sin() )
= ( A cos() ) cos(t) + ( A sin() ) sin(t)
= f(t)
with c1 = A cos(), c2 = A sin().
To reverse the process, write A =
2
c1  c2
2
and  = (1/) atan2 ( c2, c1 ).
[ Note: atan2(y, x) means “the angle whose sine and cosine are in proportion
to y, x.” A more traditional notation for the same thing is arg(x+iy), the
“argument”, or polar  coordinate, of the complex number x + iy. ]
Practice with the identity sin x = cos ( x – /2 )
5. First: is the sign in the above identity correct? Or should it be sin x = cos ( x + /2 ) ?
I think it’s correct in the header.
6. Let  be a fixed, non-zero constant. Show that the family of functions
g(t) = A cos ( ( t -  ) )
is the same as the family of functions
h(t) = A sin ( ( t -  ) ).
Just choose  in h(t) so that  in h(t) is /2 less than  in g(t).
All of the solutions to the complex-eigenvalue case are just sine waves, with
various frequencies, amplitudes and phase angles, modulated by (=multiplied
by) some exponential. The equation determines the exponential and the
frequency, and the amplitude and the phase angle are the arbitrary
constants.
3
The Trace-Determinant Plane and its Friends
1 b 
.
1
1


7. Consider the system Y’=AY where A = 
What qualitative forms of solution (sinks, sources, saddles, various kinds of spirals, etc.)
are possible? For what values / intervals of b does each occur?
The characteristic polynomial is 2 – 2 + (1-b) and its discriminant is 4b.
b>0

two real eigenvalues
…both positive if b < 1; roots 0 and 2 if b = 1; roots of
opposite signs of b > 1.
b=0

one repeated eigenvalue (namely, 1)
Also, just one eigenvector, namely (1, -1)T.
b<0

complex pair, and real part of each is 1, regardless of b.
So…
b < 0  spiral out (source)
b = 0  special case, just one eigenline
0 < b < 1  source with two eigenlines
b=1
 special case; line of equilibrium points
b > 1  saddle
4
 a 1
 . What
c
1


source
8. Now let A = 
qualitative forms are
possible? Construct an ac plane, and show which
regions in ac-space
correspond to which forms
of solution.
 a b
 . What qualitative forms are possible? Construct an a-b plane, and
 b a 
9 Now let A = 
show which regions in ab-space correspond to which forms of solution.
Trace = 2a
Determinant = a2+b2
Discriminant = T2-4D = -4b2
If b = 0, we have a repeated eigenvalue equal to a, hence one of those
one-eigenline solutions: outwards if a>0, inwards of a<0. (a=b=0 is all
constants.)
Otherwise, whether b is positive or negative we have a spiral:
out if a>0, in if a<0,
or periodic solutions (“a center”) if a=0.
10. In problem 9, for which values of (a, b) is the solution a counterclockwise spiral out ?
 x
 y
(Let Y =   . “Counterclockwise” means counterclockwise in the x-y plane with x
being measured along the horizontal axis.)
Spiraling out requires a > 0. To see whether it is counterclockwise, just test
 a b  1   a 
 0    b  . This points up

b
a

   
one point. If (x, y) = (1, 0) then 
(indicating clockwise) iff b < 0. So, the condition of the problem is met
when a > 0, b < 0.
(end)
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