Clear[Kc, vc, Ks, vs, y, Db, w, lam, L, l, solution, I1, I2, f, 
      upperl]
upperl = 10;
Kc0 = 2.18;
y0 = 1.99;
y2Ks[yy_] := 
        Sqrt[(1 + 0.5*yy)/(1 - 0.5*yy)];
y2vs[yy_] := Sqrt[1 - 0.25*yy^2];
I1[b_?NumberQ, w_] := 
      NIntegrate[(1 - Cosh[w*x])/w/
            x^2/(1 - Exp[-w*x])^(3/2), {x, b, 1}]
I2[b_?NumberQ, w_] := 
      NIntegrate[(1 - Cosh[w*x] + w*x*Sinh[w*x])/w/
            x^2/(1 - Exp[-w*x])^(3/2), {x, b, 1}]
f[KKc_, KKs_, vvc_, vvs_] := 
      1/4*(vvc/vvs)^(KKc)*((vvs/vvc)^2*(1/KKc - KKc) + 
            1/KKs - KKs)
\(\(solution = 
        NDSolve[{
\(Kc'\)[
                l] == \(-1\)/2*vc[l]/vs[l]*
                  Db[l]/Sqrt[1 - Exp[\(-w[l]\)]]*\((Kc[l]^2 - 
                      1/2*\((vc[l]/vs[l])\)^\((Kc[l] - 2)\)*\((1 + 
                            Kc[l]^2)\))\) + 
                Db[l]/4*\((vc[l]/vs[l])\)^\((Kc[l] - 1)\)*\((2 - Kc[l] - 
                      Ks[l] - y[l])\)*\((1 + Kc[l]^2)\)*\((1 + 
                      2/w[l]*Log[
                          1 + Sqrt[
                              1 - Exp[\(-w[
                                      l]\)]]])\), 
\(vc'\)[
                l] == \(-1\)/2*vc[l]^2/vs[l]*
                  Db[l]/Sqrt[1 - Exp[\(-w[l]\)]]*\((Kc[l] - 
                      1/2*\((vc[l]/vs[l])\)^\((Kc[l] - 2)\)*\((1/Kc[l] - 
                            Kc[l])\))\) + 
                Db[l]/4*\((vc[l]/vs[l])\)^\((Kc[l] - 1)\)*\((2 - Kc[l] - 
                      Ks[l] - y[l])\)*\((1/Kc[l] - Kc[l])\)*\((1 + 
                      2/w[l]*Log[
                          1 + Sqrt[
                              1 - Exp[\(-w[
                                      l]\)]]])\), 
\(Ks'\)[
                l] == \(-1\)/2*y[l]^2*Ks[l]^2 - 
                1/2*Db[l]/
                    Sqrt[1 - Exp[\(-w[l]\)]]*\((Ks[l]^2 - 
                      1/2*\((vc[l]/vs[l])\)^\((Kc[l])\)*\((1 + 
                            Ks[l]^2)\))\) + 
                Db[l]/4*\((vc[l]/vs[l])\)^\((Kc[
                        l])\)*\((\((2 - Kc[l] - Ks[l] - y[l])\)*\((1 + 
                            Ks[l]^2)\) + 4*Ks[l]^2*y[l])\)*\((1 + 
                      2/w[l]*Log[
                          1 + Sqrt[
                              1 - Exp[\(-w[
                                      l]\)]]])\), 
\(vs'\)[
                l] == \(-1\)/2*
                  Db[l]/Sqrt[1 - Exp[\(-w[l]\)]]*\((Ks[l] - 
                      1/2*\((vc[l]/vs[l])\)^\((Kc[l])\)*\((1/Ks[l] - 
                            Ks[l])\))\) + 
                Db[l]/4*\((vc[l]/vs[l])\)^\((Kc[l])\)*\((2 - Kc[l] - Ks[l] - 
                      y[l])\)*\((1/Ks[l] - Ks[l])\)*\((1 + 
                      2/w[l]*Log[1 + Sqrt[1 - Exp[\(-w[l]\)]]])\), \(y'\)[
                l] == \((2 - 2*Ks[l])\)*y[l] - 
                Db[l]/Sqrt[
                      1 - Exp[\(-w[l]\)]]*\((1 - \((vc[l]/vs[l])\)^\((Kc[
                              l])\)*\((1 - 
                            y[l]/4*\((1/Ks[l] - Ks[l])\))\))\) + 
                Db[l]/4*\((vc[l]/vs[l])\)^\((Kc[
                        l])\)*\((4*\((Ks[l] - Kc[l])\) - 
                      y[l]*\((1 - 
                            Ks[l]^2 + \((2 - Kc[l])\)*\((1/Ks[l] - 
                                  Ks[l])\))\))\)*\((1 + 
                      2/w[l]*Log[
                          1 + Sqrt[
                              1 - Exp[\(-w[
                                      l]\)]]])\), 
\(Db'\)[
                l] == \((3 - Kc[l] - Ks[l] - y[l])\)*Db[l], \(w'\)[l] == 
              w[l] + Db[l]*L[l]*
                  w[l]*\((\((1 - 
                            f[Kc[l], Ks[l], vc[l], 
                              vs[l]])\)*\(\((1 - Cosh[w[l]])\)/
                            w[l]\)/\((1 - Exp[\(-w[l]\)])\)^\((3/2)\) - 
                      f[Kc[l], Ks[l], vc[l], 
                          vs[l]]*\((2 - Kc[l] - Ks[l] - y[l])\)*
                        I1[Exp[\(-l\)], w[l]])\), 
\(lam'\)[
                l] == \(-1\)/2*Db[l]*L[l]*
                lam[l]*\((\((1 - 
                          f[Kc[l], Ks[l], vc[l], 
                            vs[l]])\)*\(\((1 - Cosh[w[l]] + 
                              w[l]*Sinh[w[l]])\)/
                          w[l]\)/\((1 - Exp[\(-w[l]\)])\)^\((3/2)\) - 
                    f[Kc[l], Ks[l], vc[l], 
                        vs[l]]*\((2 - Kc[l] - Ks[l] - y[l])\)*
                      I2[Exp[\(-l\)], w[l]])\), 
\(L'\)[
                l] == \(-L[l]\), 
Kc[0] == 
              Kc0, 
vc[0] == 
              1.0, 
Ks[0] == 
              y2Ks[y0], 
vs[0] == 
              y2vs[y0], 
y[0] == 
              y0, 
\ 
            Db[0] == 0.02, 
\ \ w[0] == 
              0.5, 
\ 
            lam[0] == 1, 
\ \ L[0] == 10}, {Kc, vc, 
            Ks, vs, y, Db, w, lam, L}, {l, 0, upperl}, 
          AccuracyGoal -> 22, PrecisionGoal -> 12, 
          MaxSteps -> 20000];\)
\)
ParametricPlot[Evaluate[{Kc[l], Db[l]} /. solution], {l, 0, upperl}, 
      PlotRange -> All, ImageSize -> 500]
Plot[Evaluate[{w[l]} /. solution], {l, 0, upperl}, 
      PlotRange -> All]
Plot[
      Evaluate[{Ks[l], Sqrt[(1 + y[l]/2)/(1 - y[l]/2)], 
            y[l]}
 /. solution], {l, 0, upperl}, 
      PlotRange -> All]
Plot[
      Evaluate[{Db[l]*L[l]*
              w[l]*((1 - 
                        1/4*(vc[l]/vs[l])^(Kc[
                                l])*((vs[l]/vc[l])^2*(1/Kc[l] - 
                                    Kc[l]) + 1/Ks[l] - Ks[l]))*(1 - 
                            Cosh[w[l]])/
                        w[l]/(1 - Exp[-w[l]])^(3/2) - 
                  1/4*(vc[l]/vs[l])^(Kc[
                          l])*((vs[l]/vc[l])^2*(1/Kc[l] - 
                              Kc[l]) + 1/Ks[l] - Ks[l])*(2 - Kc[l] - 
                        Ks[l] - y[l])*f1[Exp[-l], w[l]]), w[l]} /. 
          solution], {l, 0, upperl}, PlotRange -> All]