FE1007 Tutorial 7
March/April 2007
(1)
Consider the following first-order ordinary differential equation:
2
(
2 )
x
dy
y
x dx
y
=
+
(a)
Determine whether the equation is exact.
(b)
Multiply both sides of the equation by y
n
,
where n is an integer.
Determine the
value of n if the resulting differential equation is exact.
(c)
Solve the exact differential equation obtained in part (b).
(2)
Solve the following ODEs:
(a)
x
y
e
n y
ln
sin
0
x
x
l
dx
x
y
dy
y
⎛
⎞
⎛
⎞
+
+
+
+
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
(b)
(
)
(
)
1
3
0
y
dx
x
dy
−
+
−
=
(by two different methods).
(c)
2
'
2
xy
xy
ye
y
y
xe
+
=
−
(d)
2
2
2
2
1
1
dy
t
y
dt
t
t
=
+
+
+
Note:
1
2
2 2
2
2
2
3
1
tan
(
)
2
(
)
2
dx
x
x
c
a
a
x
a
a
x
a
−
⎛
⎞
=
+
+
⎜
⎟
⎜
⎟
+
+
⎝
⎠
∫
(3)
If the substitution
n
y
vx
=
transforms the differential equation
2
3
3
2
1
dy
x y
dx
x y
−
=
into a variable separable equation, determine the value of n.
Hence solve the
equation.

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