where skill and practice suggest a plan–of multiplying all the terms by
ϵ
a
b
t
{\displaystyle \epsilon ^{{\frac {a}{b}}t}}
, giving us:
d
y
d
t
ϵ
a
b
t
+
a
b
y
ϵ
a
b
t
=
g
b
ϵ
a
b
t
⋅
sin
2
π
n
t
{\displaystyle {\frac {dy}{dt}}\epsilon ^{{\frac {a}{b}}t}+{\frac {a}{b}}y\epsilon ^{{\frac {a}{b}}t}={\frac {g}{b}}\epsilon ^{{\frac {a}{b}}t}\cdot \sin 2\pi nt}
,
which is the same as
d
y
d
t
ϵ
a
b
t
+
y
d
(
ϵ
a
b
t
)
d
t
=
g
b
ϵ
a
b
t
⋅
sin
2
π
n
t
{\displaystyle {\frac {dy}{dt}}\epsilon ^{{\frac {a}{b}}t}+y{\frac {d(\epsilon ^{{\frac {a}{b}}t})}{dt}}={\frac {g}{b}}\epsilon ^{{\frac {a}{b}}t}\cdot \sin 2\pi nt}
;
and this being a perfect differential may be integrated thus:–since, if
d
u
d
t
=
d
y
d
t
ϵ
a
b
t
+
y
d
(
ϵ
a
b
t
)
d
t
{\displaystyle {\dfrac {du}{dt}}={\dfrac {dy}{dt}}\epsilon ^{{\frac {a}{b}}t}+y{\dfrac {d(\epsilon ^{{\frac {a}{b}}t})}{dt}}}
,
y
ϵ
a
b
t
=
g
b
∫
ϵ
a
b
t
⋅
sin
2
π
n
t
⋅
d
t
+
C
{\displaystyle y\epsilon ^{{\frac {a}{b}}t}={\frac {g}{b}}\int \epsilon ^{{\frac {a}{b}}t}\cdot \sin 2\pi nt\cdot dt+C}
,
or
y
=
g
b
ϵ
−
a
b
t
∫
ϵ
a
b
t
⋅
sin
2
π
n
t
⋅
d
t
+
C
ϵ
−
a
b
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
[A]
{\displaystyle y={\frac {g}{b}}\epsilon ^{-{\frac {a}{b}}t}\int \epsilon ^{{\frac {a}{b}}t}\cdot \sin 2\pi nt\cdot dt+C\epsilon ^{-{\frac {a}{b}}t}..............{\text{[A]}}}
The last term is obviously a term which will die out as
t
{\displaystyle t}
increases, and may be omitted. The trouble now comes in to find the integral that appears as a factor. To tackle this we resort to the device (see p. 226) of integration by parts, the general formula for which is
∫
u
d
v
=
u
v
−
∫
v
d
u
{\displaystyle \int udv=uv-\int vdu}
. For this purpose write
{
u
=
ϵ
a
b
t
;
d
v
=
sin
2
π
n
t
⋅
d
t
.
{\displaystyle {\begin{aligned}&\left\{{\begin{aligned}u&=\epsilon ^{{\frac {a}{b}}t};\\dv&=\sin 2\pi nt\cdot dt.\end{aligned}}\right.\\\end{aligned}}}
We shall then have
{
d
u
=
ϵ
a
b
t
×
a
b
d
t
;
v
=
−
1
2
π
n
cos
2
π
n
t
.
{\displaystyle {\begin{aligned}&\left\{{\begin{aligned}du&=\epsilon ^{{\frac {a}{b}}t}\times {\frac {a}{b}}\,dt;\\v&=-{\frac {1}{2\pi n}}\cos 2\pi nt.\end{aligned}}\right.\end{aligned}}}