Unformatted text preview: class, the value of theVf (t) =
As shown in solve quickly to ﬁnd ﬁxed rate investment will be Vf (t) =
2000e0.04t. From the graph below, the variable rate is better than the ﬁxed
rate for both 5 year and 10 year investments. dx x Page
thus fy (x, y )5= 1+2y .
x
So f and fy are continuous on any rectangle which excludes the line x = 0. For the IVP
with IC (ii), the E/U Theorem predicts there will be a unique solution on some interval
4. (Consider an RRSPFor thean initial investment of V0 the conditions offurther Theorem are
containing 1. with IVP with ICs (i) and (iii), dollars, where the E/U
contributions are so it makes no prediction. dollars per year. If the investment
not satisﬁed made a constant rate of k
is compounded continuously at Assignmentannual (x) = Cx/(1 given).by r(t),
(b) The general solution (from a variable 2) is y interest rate C x
thenIC (i): y (0) = 0problem for V (t), constraints on money thethe RRSP after has inﬁnitely
an initial value ) 0 = 0 ) No the amount of C. So in IVP with IC (i)
t years, is solutions. All curves corresponding to the general solution pass through (0, 0)  see
many
dV
sketch in Assignment 2 solutions. + k, V (0) = V0 . with E/U Theorem as it doesn’t apply.
= r(t)V This is consistent
dt
IC (ii): y ( 1) = 0 ) 0 = C /(1 + C ) ) C = 0. So the IVP with IC (ii) has a unique
If a person opens an with intervalthey are 20 yearsconsistent with the E/U Theorem prediction.
solution, y = 0 RRSP when of existence IR, old with an initial investment
of $1000, and (0) = 1. From (i) we knowrthat y(0)=0 for all C, so it is should
IC (iii): y the annual interest rate is (t) = 1/(t + 20) how much impossible to ﬁnd a C
they to satisfy IC (iii). There is no solution to the IVP. This is consistent with E/U Theorem
contribute per year so that they will have $500 000 at age 65? )
as it doesn’t apply.
Solution:
dV
1
dV
1
4.
=
V + k, V (0) = V0 . Standard form:
V = k.
dt
t + 20
dt
t + 20
1
1
1
Integrating factor: e t+20 dt = e ln( t+20 ) = t+20 AMATH 350 DE is the form for Theorem...
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 Winter '14
 Boundary value problem, dt, IVP

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