>>13928834Let x be a function of t, x(t).
Define the perturbed problem as cos(x(t)) = 1/sqrt(2) - pi/4 + t + x(t).
The solution is trivial for t=0 (x(0)=pi/4).
For t = pi/4 - 1/sqrt(2), you get the original problem.
Let x(t) = y(t) + pi/4.
The problem becomes [cos(y) - sin(y)]/sqrt(2) = 1/sqrt(2) + t + y
or cos(y) - sin(y) = 1 + [t + y]*sqrt(2).
Write y(t) as a power series, y1*t + y2*t^2 + y3*t^3 + ... and compute coefficients.
y1 = -sqrt(2)/[sqrt(2) + 1)
y2 = -1/[sqrt(2) + 1]^3
y3 = -[2/3 + 4sqrt(2)/3]/[sqrt(2) + 1]^5
...
Plug in t =pi/4 - 1/sqrt(2) and add pi/4.
Use whatever series acceleration method you want (Shanks would be good) for the limit of the partial sums.
You can also get a differential equation sqrt(2)y'' = (y')^3*[1 + sqrt(2)*[t + y]] and do an asymptotic expansion around 0.
