DSolve[]'s support for solving coupled differential equations is still somewhat limited,so don't be surprised if some things don't work yet.
A workaround:
ClearAll["`*"]; Remove["`*"];
eq1 =k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W1[x] +
1/2 k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 Sinh[(Sqrt[k] x)/Sqrt[
Dm]] W1[x] - (2 k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W2[x])/R0 - (
k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 Sinh[(Sqrt[k] x)/Sqrt[Dm]] W2[
x])/R0 + Dm W1''[x] == 0;
eq2 =(Dc k R0 W1[x])/(4 Dm) + (
Dc k R0 Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])] W1[x])/(2 Dm) + (
Dc k R0 Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W1[x])/(4 Dm) - (
Dc k W2[x])/(2 Dm) - (Dc k Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])] W2[x])/
Dm - (Dc k Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W2[x])/(2 Dm) +
Dc W2''[x] == 0;
EQ = eq1 /. Solve[eq2, W1[x]] /. D[Solve[eq2, W1[x]], {x, 2}] //
FullSimplify
Solution for W2(x) :
SOL = DSolve[EQ[[1]], W2[x], x] // Simplify
(*{{W2[x] ->
C[3] + x C[4] +
1/6 ((3 Dm E^((Sqrt[k] x)/Sqrt[Dm]) C[1])/k - (6 Dm C[1])/(
k + E^((Sqrt[k] x)/Sqrt[Dm]) k) + 6 x^2 C[1] - (
Dm C[2])/((1 + E^((Sqrt[k] x)/Sqrt[Dm]))^2 k) + (4 Dm C[2])/(
k + E^((Sqrt[k] x)/Sqrt[Dm]) k) - (2 Sqrt[Dm] x C[2])/Sqrt[
k] - (6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])]^2)/k + (
2 Dm C[2] Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])])/k + (
6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[
Dm])] ((3 +
4 E^((Sqrt[k] x)/Sqrt[Dm]))/(1 + E^((Sqrt[k] x)/Sqrt[
Dm]))^2 + 2 Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])]))/k + (
12 Dm C[1] PolyLog[2, -E^(((Sqrt[k] x)/Sqrt[Dm]))])/k)}}*)
Solution for W1(x) :
(Solve[eq2, W1[x]] /. SOL /. D[SOL, {x, 2}])[[1, 1, 1]] // Simplify
(*{W1[x] ->
1/(12 k R0)
Sech[(Sqrt[k] x)/(
2 Sqrt[Dm])]^2 (-3 Dm C[1] + 12 k x^2 C[1] + Dm C[2] -
4 Sqrt[Dm] Sqrt[k] x C[2] + 12 k C[3] + 12 k x C[4] +
3 Dm C[1] Cosh[(2 Sqrt[k] x)/Sqrt[Dm]] +
6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])] -
12 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])]^2 +
4 Dm C[2] Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])] +
24 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])] Log[
1 + E^((Sqrt[k] x)/Sqrt[Dm])] +
Cosh[(Sqrt[k] x)/Sqrt[
Dm]] (12 k x^2 C[1] + 3 Dm C[2] - 4 Sqrt[Dm] Sqrt[k] x C[2] +
12 k C[3] + 12 k x C[4] -
12 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])]^2 +
4 Dm C[2] Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])] +
6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[
Dm])] (3 + 4 Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])])) +
48 Dm C[1] Cosh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 PolyLog[
2, -E^(((Sqrt[k] x)/Sqrt[Dm]))] -
12 Dm C[1] Sinh[(Sqrt[k] x)/Sqrt[Dm]] -
3 Dm C[2] Sinh[(Sqrt[k] x)/Sqrt[Dm]] -
18 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])] Sinh[(Sqrt[k] x)/Sqrt[
Dm]])}*)