Unformatted text preview: has one eigenvalue,
(a) compare .your answer with part (a). , of multiplicity three with one
Show 0
x < 0.
(b) linearly independent eigenvector,x), y2 (x)Ay ﬁnd (one solution of the
Use general solution of the y1 ( v1 y = are linearly dependent on
deﬁnition show that
6. Find thethe deﬁnition show that systemy.2 (Hence linearly where
(c) Use the
y 1 ( x) ,
x) are + F x) independent on
system.
x < 0.
IR.
21
4xe x
(b) Find two more independent solutions) ) described .
A=
(c) Use the deﬁnition show that ,yand ,F(xxasare linearly in class. Verify
( x) y 2 ( =
independent on
1
2
(d) Show that W (y1 (x), y2 (x)) = 1 for all x IR. 0
0
that
IR. the three solutions are linearly independent. Hence give the
7. Consider a solution of )withxconstant supply SIR. 50 and demand curve
=
(d) general commodity the2system.0 for all x
Show that W (y1 (x , y ( )) =
whose derivative is given by
dD
= 2D 5p + 10
dt
where D is the demand and p is the price. The rate of increase of the
price is equal to the rate of decrease of the quantity.
(a) Write down a system of di erential equations for D(t) and p(t) that
models the situation described above....
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 Fall '12
 DavidHamsworth
 Linear Algebra, Boundary value problem, Eigenvalue, eigenvector and eigenspace, Generalized eigenvector, di erential equations, andxgeneralized eigenvector u. deﬁnition

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