Off[General::spell]; Off[General::spelll];

\(\(\(
\)\(
\)\(
\)\(\(pr = 400;\)

    \(o = 300;\)



    \(hb = SetPrecision[1, pr];\)

    \(m = SetPrecision[1, pr];\)


    \(Clear[k, l, Eo];\)

    \(G[y_] := \(-\((1/4)\)\)*y;\)

    \(K[x_] := Sum[k[n]*x^n, {n, 0, o + 1}];\)

    \(L[x_] := Sum[l[n]*x^n, {n, 0, o + 1}];\)

    \(V[y_] = \((1/2)\)*y^2;\)


    \(eq1 = hb*D[K[y], y] + L[y] + G[y]*K[y];\)

    \(eq2 = 
        hb*D[L[y], y] + K[y]*\((V[y] - Eo)\)*2*m + 
          L[y]*G[y];\)

    \(alleq1 = Series[Expand[eq1], {y, 0, o}];\)

    \(alleq2 = 
        Series[Expand[eq2], {y, 0, 
            o}];\)



    \(Do[
eq1ao[ll] = Expand[alleq1[\([3, ll + 1]\)]]; 
        eq2ao[ll] = Expand[alleq2[\([3, ll + 1]\)]]
, {ll, 
          0, o}];\)


 (*even\ parity\ with\ respect\ to\ symmetric\ origin*) 

    k[0] = 1; l[0] = 0;

    \(Do[sol1 = Solve[Expand[eq1ao[ll]] == 0, k[ll + 1]]; 
        k[ll + 1] = k[ll + 1] /. sol1[\([1]\)]; 
sol2 = 
          Solve[Expand[eq2ao[ll]] == 0, 
            l[ll + 1]]; 
l[ll + 1] = 
          l[ll + 1] /. sol2[\([1]\)], {ll, 0, o}];\)

    \)\)\)

Kp = Expand[K[5]]; 
    Plot[Log[Abs[Kp]], {Eo, 0, 10}]

\(\(\(
\)\(
\)\(
\)\(dKp = D[Kp, Eo]; \ eg = SetPrecision[1/2, pr];

    \(n = 0;\)

    \(While[Abs[Kp /. Eo -> eg] > 10^\((\(-40\))\), 
        eg = eg - \((\((Kp/dKp)\) /. Eo -> eg)\); n = n + 1; 
        Print[N[eg, 40]]];\)\)\)\)

dKp = D[Kp, Eo];  eg = SetPrecision[1.6, pr];
n = 0;
While[Abs[Kp /. Eo -> eg] > 10^(-40), 
        eg = eg - ((Kp/dKp) /. Eo -> eg); n = n + 1; 
        Print[N[eg, 40]]];