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Quiz for June 5, 2012
The quiz is worth 5 points.
Remove EVERYTHING from your desk except
this quiz and a pen or pencil.
SHOW your work. Express your work in a neat
and coherent manner.
BOX
your answer.
1.
Solve
yy
′
+
x
=
r
x
2
+
y
2
. Express your answer in the form
y
(
x
)
. Check
your answer.
This is a homogeneous problem. Divide both sides by
x
to write the problem as
y
x
y
′
+ 1 =
R
1 +
p
y
x
P
2
.
Let
v
=
y
x
. In other words,
xv
=
y
. Take the derivative with respect to
x
to see
that
xv
′
+
v
=
y
′
. We must solve
v
(
xv
′
+
v
) + 1 =
r
1 +
v
2
.
We must solve
xv
dv
dx
=
r
1 +
v
2

v
2

1
.
We must solve
v
dv
√
1 +
v
2

v
2

1
=
dx
x
.
Integrate both sides. Let
w
= 1 +
v
2
. It follows that
dw
= 2
vdv
. We must solve
1
2
i
dw
√
w

w
= ln

x

+
C.
We have
ln

x

+
C
=
1
2
i
dw
√
w
(1

√
w
)
.
Let
u
=
√
w
. We have
du
=
1
2
w
−
1
/
2
dw
.
ln

x

+
C
=
i
du
1

u
=

ln

1

u

=

ln

1

√
w

=

ln

1

r
1 +
v
2

=

ln
v
v
v
v
v
1

R
1 +
p
y
x
P
2
v
v
v
v
v
=

ln
v
v
v
v
v
x

r
x
2
+
y
2
x
v
v
v
v
v
=

ln
v
v
v
x
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 Fall '11
 Staff
 Differential Equations, Equations, Trigraph, coherent, Natural logarithm, ln x

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