Differential Equations
Test 1 Solutions
Answer FOUR questions. Show all working clearly.
1. Solve the initial value problem
d
y
d
x
=
y
2
+ 2
xy
2
,
y
(0) = 1
.
2. Solve the initial value problem
d
y
d
x

y
= 2
xe
x
,
y
(0) = 1
.
3. Find the value of
A
for which the differential equation
(2
x
+
ye
xy
)d
x
+
Axe
xy
d
y
= 0
is exact and solve it with this value.
4. Solve the differential equation
d
y
d
x
+
y
x
=
y
2
x
2
.
5. A population
N
has intrinsic growth rate
r >
0 and is harvested at constant
rate
h >
0 so that
d
N
d
t
=
rN

h.
Determine
N
as a function of
t
and its initial value
N
0
>
0. Show that there
is a threshold value
k
such that if
h < k
then
N
grows without bound while
if
h > k
then the population is driven to extinction.
1
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Solutions
:
1. The given DE
d
y
d
x
= (1 + 2
x
)
y
2
is separable. Separate and integrate:

1
y
=
d
y
y
2
=
(1 + 2
x
)d
x
=
x
+
x
2
+
k
and impose the initial condition
y
(0) = 1 to find
k
=

1. Conclusion:
y
=
1
1

x

x
2
.
2. The given DE is linear (already normalized) with integrating factor
exp(

1d
x
) =
e

x
so that
d
d
x
(
ye

x
) = 2
x
and therefore
ye

x
=
x
2
+
k.
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 Fall '08
 TUNCER
 Trigraph, Constant of integration, Boundary value problem, dx, initial condition

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