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Unformatted text preview: be rewritten as
e y = ex , and so we ﬁnd that
dx a) For y=e 2x (ex + C ) = e x + Ce 2x . We can see that y = e x is an exceptional solution, and all other solutions
approach it as x ! 1, and are repelled by it as x ! 1.
Next, observe that y 0 = 0 if and only if y = 1 e x (this is our horizontal
2
isocline). The solutions must look something like this: AMATH 350 Assignment #3  Winter 2013 Page 3 dy
b) For x
= 2y + x3 , we need to start by putting the equation in standard
dx
form:
dy
2
y = x2 .
dx x
R 2 An integrating factor is µ(x) = e x dx = e 2 ln x = 1/x2 . Incorporating
this, we obtain
1 dy
2
y=1
2 dx
x
x3
which must be the same equation as Therefore
and so dy ⇣ y ⌘
= 1.
dx x2
y
=x+C
x2
y = x3 + Cx2 . Now, y = x3 is an exceptional solution. All other solutions approach this
as x ! 0 and diverge from it as x ! ±1.
x3
The horizontal isocline is y =
. Finally, it might help to recognize that
2
every solution passes through the origin. This is unusual, since solution
curves usually cannot cross, but in this particular example the conditions
of the Existence and Uniqueness Theorem are not met on the y axis.
Putting our observations together, we arrive at the following sketch: AMATH 350 Assignment #3  Winter 2013 Page 4 3. (Interest Rate Problem)
a) An initial value problem for the interest rate is
dr
= 0.0015,
r(0) = 0.05
dt
Since y = xv , the general solution of the original DE is
x
Solve this y = ln the+ C.
+ ln to ﬁnd x variable interest rate r(t).
y
Solution: x
dr the original
r(0) = 0.05 .
6. (a) Sincefor = xv , the general solution of = 0.0015 DE is
IVP y interest rate r(t):
x
y
dt
+ ln
= ln x + C.
initial rate is 5%
y
xZ
rate decreases 0.15% per year
Z
Solve DE: dr =
0.0015 dt )dr = 0.0015t + C .
r
= 0.0015
r(0) = 0.05 .
6. (a) IVP for interest rate r(t):
Apply IC: r(0) = 0.05 ) C = 0.05.
dt
initial rate is 5%
Interest rate at time t: r(t) = decreases0015t. per year
rate 0.05 0. 0.15%
Z
Z
Since (t= solve general0015 investment 0(dollars) is
the
(b) Set upyandxv ,dr =value 0ofsolutionproblem for theDE at. time t investment,
Let V )
. va...
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