the complementary function of (D⁴-a⁴)y=0 is
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Answered by
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Step-by-step explanation:
With linear operator
D
=
d
d
x
, this is:
(
D
4
−
4
)
y
=
0
(
D
2
−
2
)
(
D
2
+
2
)
y
=
0
(
D
−
√
2
)
(
D
+
√
2
)
(
D
−
i
√
2
)
(
D
+
i
√
2
)
y
=
0
Each of these factors has a simple solution, eg:
(
D
−
√
2
)
y
=
0
⇒
y
'
=
√
2
y
⇒
y
=
α
e
√
2
x
(
D
−
i
√
2
)
y
=
0
⇒
y
'
=
i
√
2
y
⇒
y
=
β
e
i
√
2
x
The complete solution is the superposition of individual solutions:
⇒
y
(
x
)
=
A
e
√
2
x
+
B
e
−
√
2
x
+
C
e
i
√
2
x
+
D
e
−
i
√
2
x
C
e
i
√
2
x
+
D
e
−
i
√
2
x
can also be re-written using the Euler formula , arriving at:
y
=
A
e
√
2
x
+
B
e
−
√
2
x
+
C
'
cos
(
√
2
x
)
+
D
'
sin
(
√
2
x
)
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