1.Method one:
In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
F[x_,A_,B_]:=(NSolveValues[y*Sinh[A*Log[y]+B]==x,y,Reals][[1]])
In[]:=
Plot[F[x,1,1],{x,-0.18,3}]
Out[]=
2. Method two:
In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
A=1;B=1;ContourPlot[y*Sinh[A*Log[y]+B]==x,{x,-0.5,3},{y,-2,2},FrameFalse,AxesTrue,AxesLabel{x,y}]
Out[]=
3. Method tree:
ClearAll["`*"];Remove["`*"];
In[]:=
A=1;B=1;ParametricPlot[{x*Sinh[A*Log[x]+B],x},{x,0,3/2}]
Out[]=
4. Method four:
In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
A=1;B=1;G[x_]:=Evaluate[(InverseSeries[Series[x*Sinh[A*Log[x]+B],{x,1,20}]]//Normal)](*Seriesaproximation!*)
In[]:=
G[x]
Out[]=
1-
883631595
20
(x-Sinh[1])
262144
20
(Cosh[1]+Sinh[1])
+
119409675
19
(x-Sinh[1])
65536
19
(Cosh[1]+Sinh[1])
-
64822395
18
(x-Sinh[1])
65536
18
(Cosh[1]+Sinh[1])
+
17678835
17
(x-Sinh[1])
32768
17
(Cosh[1]+Sinh[1])
-
9694845
16
(x-Sinh[1])
32768
16
(Cosh[1]+Sinh[1])
+
334305
15
(x-Sinh[1])
2048
15
(Cosh[1]+Sinh[1])
-
185725
14
(x-Sinh[1])
2048
14
(Cosh[1]+Sinh[1])
+
52003
13
(x-Sinh[1])
1024
13
(Cosh[1]+Sinh[1])
-
29393
12
(x-Sinh[1])
1024
12
(Cosh[1]+Sinh[1])
+
4199
11
(x-Sinh[1])
256
11
(Cosh[1]+Sinh[1])
-
2431
10
(x-Sinh[1])
256
10
(Cosh[1]+Sinh[1])
+
715
9
(x-Sinh[1])
128
9
(Cosh[1]+Sinh[1])
-
429
8
(x-Sinh[1])
128
8
(Cosh[1]+Sinh[1])
+
33
7
(x-Sinh[1])
16
7
(Cosh[1]+Sinh[1])
-
21
6
(x-Sinh[1])
16
6
(Cosh[1]+Sinh[1])
+
7
5
(x-Sinh[1])
8
5
(Cosh[1]+Sinh[1])
-
5
4
(x-Sinh[1])
8
4
(Cosh[1]+Sinh[1])
+
3
(x-Sinh[1])
2
3
(Cosh[1]+Sinh[1])
-
2
(x-Sinh[1])
2
2
(Cosh[1]+Sinh[1])
+
x-Sinh[1]
Cosh[1]+Sinh[1]
In[]:=
Plot[G[x],{x,-1,3.2}]
Out[]=
(*---------------------------------------------------------------------*)
In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
A=1;B=1;AsymptoticSolve[y*Sinh[A*Log[y]+B]-x==0,{y,0},{x,1,5}](*Can'tsolve*)
Out[]=
AsymptoticSolve[-x+ySinh[1+Log[y]]0,{y,0},{x,1,5}]
AsymptoticSolve[Sin[xy]-y^2+x0,{y,0},{x,0,5}](*Worksfineforanotherexample.*)
Out[]=
y-
x
+
x
2
-
3/2
x
8
+
5/2
x
128
-
7/2
x
1024
-
4
x
12
+
4101
9/2
x
32768
-
5
x
12
,y
x
+
x
2
+
3/2
x
8
-
5/2
x
128
+
7/2
x
1024
-
4
x
12
-
4101
9/2
x
32768
-
5
x
12

(*---------------------------------------------------------------------*)
In[]:=
ClearAll["`*"];Remove["`*"];
Solve[y*Sinh[A*Log[y]+B]==x,y](*Can'tsolve*)
Solve
:This system cannot be solved with the methods available to Solve. Try Reduce or FindInstance instead.
Out[]=
Solve[ySinh[B+ALog[y]]x,y]
Reduce[y*Sinh[2*Log[y]+2]==x,y](*Can'tsolve*)
Reduce
:This system cannot be solved with the methods available to Reduce.
Out[]=
Reduce[ySinh[2+2Log[y]]x,y]