Problems with Dsolve[] and simple function

Question

I'm trying to solve the following very simple differential equation, but it seems Mathematica cannot give me an answer,

FullSimplify[DSolve[{y''[x] == 2 A Sinh[B y[x]], y[0] == W, y[L] == W}, y[x], x]]

Any thoughts? Thanks!

Answer 1

When I try I get warnings about not being able to solve with the given conditions. So for a start, try solving without conditions.

sol = DSolve[{y''[x] == 2 A Sinh[B y[x]]}, y[x], x] // Flatten

You get two solutions involving JacobiAmplitude. Setting the two solutions:

y1[x_] = y[x] /. sol[[1]] /. {C[1] -> c1, C[2] -> c2}

y2[x_] = y[x] /. sol[[2]] /. {C[1] -> c1, C[2] -> c2}

Theoretically you can solve for c1 and c2 by plugging in your conditions, but you get transcendental equations that MMa cannot solve. If you provide values for A, B and W, you can probably find numerical solutions for the c's with FindRoot or NSolve.

Answer 2

Clear["Global`*"]

Use a numeric approach

L = 5;

eqns = {y''[x] == 2 A Sinh[B y[x]], y[0] == W, y[L] == W};

sol = ParametricNDSolve[eqns, y, {x, 0, L}, {A, B, W}];

y[A, B, W][x] /. {A -> 1, B -> 1, W -> 3/4, x -> 3} /. sol

(* 0.0544074 *)

Plot3D[y[1, 1, W][x] /. sol, {x, 0, L}, {W, -1, 1},
 AxesLabel -> (Style[#, 14, Bold] & /@ {"x", "W", "y"}),
 ClippingStyle -> None,
 ColorFunction -> "TemperatureMap",
 PlotLegends -> Automatic]