Now I tried to change the interval as follows:

A0 = 2.0377638272727268`;
A1 = -7.105521894545453`;
A2 = 9.234000147272726`;
A3 = -5.302489919999999`;
A4 = 1.1362478399999998`;
h0 = 45.5;
\[Sigma]M = 0.00592251;
\[Lambda]1 = 1.0253896074561006`;
\[Lambda]2 = 1.3079437258774012`;

f1 = A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3;
b = h0^2/12 (5 A1 + 8 A2 y[x] + 9 A3 y[x]^2 + 8 A4 y[x]^3)/y[x]^6;
g = -(h0^2/12) (A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3)/y[x]^5;
eqn = f1 + b y'[x]^2 + g y''[x] - \[Sigma]M;

LA = 0;
LB = 15;

sol = NDSolve[{
     eqn == 0,
     y[LA] == \[Lambda]1,
     y[LB] == \[Lambda]2
     }, y, {x, 0, 15}, 
     Method -> {"Shooting", "StartingInitialConditions" -> {y[LB] == \[Lambda]2, y'[LB] == 0}}
     ]

Show[
     Plot[y[x] /. sol, {x, 0, LB}, PlotRange -> {Automatic, {1, 1.32}}, Frame -> True],
     Graphics[{
     Red,
     Point[{{LA, \[Lambda]1}, {LB, \[Lambda]2}}]
      }]
    ]

If I set LB different from 15, something strange happens.. Moreover, even if LB = 15, the boundary condition in x = 0 is not satisfied. Where is the problem?

Now I tried to change the interval as follows:

A0 = 2.0377638272727268`;
A1 = -7.105521894545453`;
A2 = 9.234000147272726`;
A3 = -5.302489919999999`;
A4 = 1.1362478399999998`;
h0 = 45.5;
σM = 0.00592251;
λ1 = 1.0253896074561006`;
λ2 = 1.3079437258774012`;

f1 = A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3;
b = h0^2/12 (5 A1 + 8 A2 y[x] + 9 A3 y[x]^2 + 8 A4 y[x]^3)/y[x]^6;
g = -(h0^2/12) (A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3)/y[x]^5;
eqn = f1 + b y'[x]^2 + g y''[x] - σM;

LA = 0;
LB = 15;

sol = NDSolve[{
     eqn == 0,
     y[LA] == λ1,
     y[LB] == λ2
     }, y, {x, 0, 15}, 
     Method -> {"Shooting", "StartingInitialConditions" -> {y[LB] == λ2, y'[LB] == 0}}
     ]

Show[
     Plot[y[x] /. sol, {x, 0, LB}, PlotRange -> {Automatic, {1, 1.32}}, Frame -> True],
     Graphics[{
     Red,
     Point[{{LA, λ1}, {LB, λ2}}]
      }]
    ]

If I set LB different from 15, something strange happens.. Moreover, even if LB = 15, the boundary condition in x = 0 is not satisfied. Where is the problem?

Now I tried to change the interval as follows:

A0 = 2.0377638272727268`;
A1 = -7.105521894545453`;
A2 = 9.234000147272726`;
A3 = -5.302489919999999`;
A4 = 1.1362478399999998`;
h0 = 45.5;
\[Sigma]M = 0.00592251;
\[Lambda]1 = 1.0253896074561006`;
\[Lambda]2 = 1.3079437258774012`;

f1 = A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3;
b = h0^2/12 (5 A1 + 8 A2 y[x] + 9 A3 y[x]^2 + 8 A4 y[x]^3)/y[x]^6;
g = -(h0^2/12) (A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3)/y[x]^5;
eqn = f1 + b y'[x]^2 + g y''[x] - \[Sigma]M;

LA = 0;
LB = 15;

sol = NDSolve[{
     eqn == 0,
     y[LA] == \[Lambda]1,
     y[LB] == \[Lambda]2
     }, y, {x, 0, 15}, 
     Method -> {"Shooting", "StartingInitialConditions" -> {y[LB] == \[Lambda]2, y'[LB] == 0}}
     ]

If I set LB different from 15, something strange happens.. Moreover, even if LB = 15, the boundary condition in x = 0 is not satisfied. Where is the problem?

Now I tried to change the interval as follows:

A0 = 2.0377638272727268`;
A1 = -7.105521894545453`;
A2 = 9.234000147272726`;
A3 = -5.302489919999999`;
A4 = 1.1362478399999998`;
h0 = 45.5;
\[Sigma]M = 0.00592251;
\[Lambda]1 = 1.0253896074561006`;
\[Lambda]2 = 1.3079437258774012`;

f1 = A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3;
b = h0^2/12 (5 A1 + 8 A2 y[x] + 9 A3 y[x]^2 + 8 A4 y[x]^3)/y[x]^6;
g = -(h0^2/12) (A1 + 2 A2 y[x] + 3 A3 y[x]^2 + 4 A4 y[x]^3)/y[x]^5;
eqn = f1 + b y'[x]^2 + g y''[x] - \[Sigma]M;

LA = 0;
LB = 15;

sol = NDSolve[{
     eqn == 0,
     y[LA] == \[Lambda]1,
     y[LB] == \[Lambda]2
     }, y, {x, 0, 15}, 
     Method -> {"Shooting", "StartingInitialConditions" -> {y[LB] == \[Lambda]2, y'[LB] == 0}}
     ] 

Show[
     Plot[y[x] /. sol, {x, 0, LB}, PlotRange -> {Automatic, {1, 1.32}}, Frame -> True],
     Graphics[{
     Red,
     Point[{{LA, \[Lambda]1}, {LB, \[Lambda]2}}]
      }]
    ]

If I set LB different from 15, something strange happens.. Moreover, even if LB = 15, the boundary condition in x = 0 is not satisfied. Where is the problem?