# Solving BVP using shooting method and plotting the result

I am not able to obtain any plot with the following code.

P = ParametricNDSolve[{(ϵ Cos[x] (κ (1 + ϵ Sin[x]) -
Tanh[κ (1 + ϵ Sin[x])] - κ (1 + ϵ Sin[x]) Tanh[κ (1 + ϵ Sin[
x])]^2 + κ (1 + ϵ Sin[x])^4 Derivative[1][
p][x]))/(κ (1 + ϵ Sin[x])^2) +
2/3 (1 + ϵ Sin[x])^3 (p^′′)[x] == 0, p[0] == 0, p[1] == 0}, p, {x, 0, 1}, {ϵ, κ},
Method -> {"Shooting", "StartingInitialConditions" -> {p[0] == 0, p'[0] == 1}}]

Plot[P[0.8, 100][x], {x, 0, 1}]


What is wrong with my code?

You have syntax errors in specifying derivatives. Further,"ParametricNDSolve" returns a rules, not a function.

Here is the corrected code:

sol = p /.
ParametricNDSolve[{(ϵ Cos[
x] (κ (1 + ϵ Sin[x]) -
Tanh[κ (1 + ϵ Sin[
x])] - κ (1 + ϵ Sin[
x]) Tanh[κ (1 + ϵ Sin[
x])]^2 + κ (1 + ϵ Sin[x])^4 p' [
x]))/(κ (1 + ϵ Sin[x])^2) +
2/3 (1 + ϵ Sin[x])^3 p''[x] == 0, p[0] == 0,
p[1] == 0}, p, {x, 0, 1}, {ϵ, κ},
Method -> {"Shooting",
"StartingInitialConditions" -> {p[0] == 0, p'[0] == 1}}]

Plot[sol[0.8, 100][x], {x, 0, 1}]


"ShootingMethod" isn't necessary, try

P = ParametricNDSolveValue[{(\[Epsilon] Cos[
x] (\[Kappa] (1 + \[Epsilon] Sin[x]) -
Tanh[\[Kappa] (1 + \[Epsilon] Sin[
x])] - \[Kappa] (1 + \[Epsilon] Sin[
x]) Tanh[\[Kappa] (1 + \[Epsilon] Sin[
x])]^2 + \[Kappa] (1 + \[Epsilon] Sin[x])^4 p'[
x]))/(\[Kappa] (1 + \[Epsilon] Sin[x])^2) +
2/3 (1 + \[Epsilon] Sin[x])^3 p''[x] == 0, p[0] == 0, p[1] == 0},
p, {x, 0, 1}, {\[Epsilon], \[Kappa]}]
Plot[P[0.8, 100][x], {x, 0, 1}]