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Math 101 Chapter 3: Differentiation
3.36 A differential equation for hyperbolic functions
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3.37 Solution of differential equation for hyperbolic functions
We have
f
′
(
x
)
=
β
A
sinh
β
x
+
B
cosh
β
x
f^{\prime}(x)=\beta A\sinh\beta x+B\cosh\beta x
f
′′
(
x
)
=
β
2
A
cosh
β
x
+
B
β
sinh
β
x
f^{\prime\prime}(x)=\beta^{2}A\cosh\beta x+{{B}{\beta}}\sinh\beta x
so
f
(
0
)
=
A
f(0)=A
,
f
′
(
0
)
=
B
f^{\prime}(0)=B
and
f
′′
(
x
)
=
β
2
f
(
x
)
=
k
m
f
(
x
)
.
f^{\prime\prime}(x)=\beta^{2}f(x)={{k}\over{m}}f(x).