Using Mathematica, I want to solve an ODE of an Interpolating Function. First I solve the ODE obtaining the function y[s] (or Y[s])
aa = .4; sol = NDSolve[{-4 + 4/y[s]^2 - y[s]^2 + y[s]^4 + y'[s]^2 == 0, y[0] == 1.207}, y[s], {s, 0, aa}];
ParametricPlot [Evaluate[{s, y[s]} /. sol[[2]]], {s, 0, aa}, PlotRange -> All]
Y[s_] = y[s] /. sol[[2]];
(with the function Y[s] I can plot the solution y[s])
My problem is solving the ODE w'[s]= Sqrt[y[s]^2 + 2]
with initial condition w[0] = 0.
My first attempt has been
w1 = NDSolve[{w'[s] == Sqrt[Y[s]^2 + 2], w[0] == 0}, w[s], {s, 0, aa}]
Next, I put W[s_] = w[s] /. w1[[1]]
and I plot the function W[s] by means of Plot[W[s], {s, 0, aa}]
. However, I think that it is wrong because the plot of W[s] seems to be a straight-line. In fact, I hope that Mathematica solves the ODE for w[s] by taking only the first value for y[s], namely, y[0]=1.207
and then solves the ODE w'[s]=Sqrt[1.207^2+2], w[0]=0
(whose solution is a linear function). I have computed the last ODE:
p1 = NDSolve[{p'[s] == Sqrt[1.207^2 + 2], p[0] == 0}, p[s], {s, 0, aa}]
and the plots of both functions p[s] and w[s] are 'very similar'. Please, help.