In[]:=
DSolve[α''[x]+ρSin[α[x]]==0,α[x],x]
Solve
:Inverse functions are being used by Solve​, so some solutions may not be found; use Reduce for complete solution information.
Out[]=
α[x]-2JacobiAmplitude
1
2
(2ρ+

1
)
2
(x+

2
)
,
4ρ
2ρ+

1
,α[x]2JacobiAmplitude
1
2
(2ρ+

1
)
2
(x+

2
)
,
4ρ
2ρ+

1

In[]:=
JacobiAmplitude
x
2
(2ρ+

1
)
,
4ρ
2ρ+

1
+O[x]^2
Out[]=
1
2
2ρ+

1
x+
2
O[x]
In[]:=
JacobiAmplitude
x
2
(2ρ1+

1
)
,
4ρ1
2ρ1+

1
-JacobiAmplitude
x
2
(2ρ2+

2
)
,
4ρ2
2ρ2+

2
+O[x]^2
Out[]=
1
2

2ρ1+

1
-
2ρ2+

2
x+
2
O[x]
In[]:=
sub={

1
->a^2-2ρ1,

2
->a^2-2ρ2}
Out[]=
{

1

2
a
-2ρ1,

2

2
a
-2ρ2}
In[]:=
JacobiAmplitude
x-ξ2
2
(2ρ1+

1
)
,
4ρ1
2ρ1+

1
/.sub
Out[]=
JacobiAmplitude
1
2
2
a
(x-ξ2),
4ρ1
2
a

In[]:=
JacobiAmplitude
x-ξ2
2
(2ρ2+

2
)
,
4ρ2
2ρ2+

2
/.sub
Out[]=
JacobiAmplitude
1
2
2
a
(x-ξ2),
4ρ2
2
a

In[]:=
F[a_,r_,ξ_]:=JacobiAmplitudeξ2a,
r
a^2ξ
-JacobiAmplitude
(1-ξ)
2
a,
-r
a^2(1-ξ)

In[]:=
FindRoot[F[a,1000,0.1],{a,-10,-0.1}]
Out[]=
{a-2.88328}
In[]:=
FindRoot[F[0.0001,r,0.55],{r,40}]
Out[]=
{r4.79313}
In[]:=
r=.;
In[]:=
4IntegrateDJacobiAmplitude(x-ξ)2a,
r
a^2ξ
,x^2,{x,0,ξ}
Out[]=
2aJacobiEpsilon
aξ
2
,
r
2
a
ξ

In[]:=
4IntegrateDJacobiAmplitude(x-ξ)2a,-
r
a^2(1-ξ)
,x^2,{x,ξ,1}
Out[]=
2aJacobiEpsilon
1
2
a(1-ξ),
r
2
a
(-1+ξ)

In[]:=
2aJacobiEpsilon
aξ
2
,
r
2
a
ξ
+JacobiEpsilon
1
2
a(1-ξ),
r
2
a
(-1+ξ)

Out[]=
2aJacobiEpsilon
1
2
a(1-ξ),
r
2
a
(-1+ξ)
+JacobiEpsilon
aξ
2
,
r
2
a
ξ

In[]:=
​
In[]:=
IntegrateExpI2JacobiAmplitude(x-ξ)2a,
r
a^2ξ
,{x,0,ξ}
Out[]=
ξ
∫
0
2JacobiAmplitude
1
2
a(x-ξ),
r
2
a
ξ


x
In[]:=
2DJacobiAmplitude(x-ξ)2a,
r
a^2ξ
,x/.x->0
Out[]=
aJacobiDN
aξ
2
,
r
2
a
ξ

In[]:=
2DJacobiAmplitude(x-ξ)2a,
r
a^2ξ
,x/.x->ξ
Out[]=
a
In[]:=
a^2JacobiDN
aξ
2
,
r
2
a
ξ

Out[]=
2
a
JacobiDN
aξ
2
,
r
2
a
ξ

In[]:=
Plot[F[a,1000,0.1],{a,-10,10}]
Out[]=
-10
-5
5
10
-3
-2
-1
1
2
3
In[]:=
Plot[F[a,100,0.1],{a,-10,10}]
Out[]=
-10
-5
5
10
-4
-2
2
4
In[]:=
Plot[F[a,10,0.1],{a,-10,10}]
Out[]=
-10
-5
5
10
-4
-2
2
4
In[]:=
Plot[{F[a,10,0.1],F[a,100,0.1],F[a,1000,0.1],F[a,10000,0.1]},{a,-10,10},​​PlotLabel->"F[a,r,0.1]",AxesLabel->{"a",None},​​PlotLegends->{"r=10","r=100","r=1000","r=10000"}]
Out[]=
r=10
r=100
r=1000
r=10000
In[]:=
​
In[]:=
′′
f
1,2
(ξ)+
λ
1,2
f
1,2
(ξ)
-ϵ
τ
f
1,2
(ξ);​​
λ
1,2

r
Δ
,
r
Δ-1

Out[]=
λ
1,2

r
Δ
,
r
-1+Δ

In[]:=
​​(*Thesolutionmatchingwithfirstderivativeat$\xi=\xi_2$reads(uptoirrelevantcommonnormalizationfactor)*)​​
f
1
(ϵ,ξ)cos(
K
1
(ξ-
ξ
2
))+
B
K
1
sin(
K
1
(ξ-
ξ
2
));​​
f
2
(ϵ,ξ)cos(
K
2
(ξ-
ξ
2
))+
B
K
2
sin(
K
2
(ξ-
ξ
2
));\[NewLine]
K
1,2

λ
1,2
+
ϵ
τ
​​