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1
FirstOrder Equations: Linear Equations Problem Set
Solve the following linear firstorder differential equations.
1.
'2
2
2
t
yy
t
e
−=
Comparing
this
equation
with
the
general
linear
firstorder
equation,
()
dy
pty gt
dx
+=
, we can see that,
2
pt
= −
,
22
t
gt
te
=
. Therefore, the
integrating factor is:
2
2
p t dt
dt
t
µµ
µ
−
−
∫∫
=⇒
=
⇒
=
Multiplying both side of the equation by the integrating factor, we get:
3
2'
2
2 2
2
2 '
2
2
2
2
2
2
2(
)
(
)
(
)
3
tt
t
t
t
t
t
t
dt
ey ey e t
e
ey t
d
t ey
C
dt
−−
−
−
−
−
−
−
=
⇒
=
⇒
=
⇒=
+
∫
3
2
3
t
t
ye
C
=+
2.
(1
)
4
)
ty t
y
t
−
++
=
+
'
41
)
4
)
11
t
y
t
y
y
−
=
+⇒
+
=
+
+
Comparing
this
equation
with
the
general
linear
firstorder
equation,
dy
dx
, we can see that:
2
4
1
t
t
=
+
. Therefore, the
integrating factor is:
ptd
t
∫
2
2
4
2ln(1
)
ln(1
)
2
2
1
)
t
dt
t
e
e
t
+
∫
==
=
=
+
Multiplying both side of the equation by the integrating factor, we get:
22 '
2
2
'
2
2
)
4 (1
)
1
((1
)
)
1
)
)
ty
t
t
y
t
t
t
d
t
+
+ +
=+⇒ +
=+⇒+
= +
∫
⇒
1
)
tan
( )
tC
−
+
1
tan
( )
)
y
t
−
+
∴
=
+
3.
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 Fall '10
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