Code:

(* solve equation numerically in u *)
sol = NDSolve[{Derivative[2][ua][x] -
x^2*ua[x] == 0, ua[0] == 1,
Derivative[1][ua][0] == 2}, ua,
{x, 0, 5}]
p1 = Plot[ua[x] /. sol, {x, 0, 3},
PlotStyle -> Blue]
(* plot u as the power series *)
kmax = 25;
a0 = 1;
a1 = 2;
ub[x_] :=
a0*(1 + Sum[x^(4*k)/Product[
(3 + 4*j)*(4 + 4*j),
{j, 0, k - 1}], {k, 1, kmax}]) +
a1*(x + Sum[x^(4*k + 1)/Product[
(4 + 4*j)*(5 + 4*j),
{j, 0, k - 1}], {k, 1, kmax}]);
p2 = Plot[ub[x], {x, 0, 3},
PlotStyle -> Red]
netinu = Show[{p1, p2}]
(* solve equation in y numerically *)
sol2 = NDSolve[{Derivative[1][yb][x] +
yb[x]^2 - x^2 == 0, yb[0] == 2},
yb, {x, 0, 3}]
p3 = Plot[yb[x] /. sol2, {x, 0, 3},
PlotStyle -> Blue]
(* plot equation in y from power series in u *)
ya[x_] = D[ub[x], x]/ub[x];
p4 = Plot[ya[x], {x, 0, 3},
PlotStyle -> Red]
netiny = Show[{p4, p3}]