The functions C = C(x) and S = S(x) satisfy:
(1a) sqrt(C) - sqrt(S) = 1.
(1b) C'*sqrt(S) = S'*sqrt(C) = 72*x.
(1c) C' = 72*x/sqrt(S).
(1d) S' = 72*x/sqrt(C).
Integrals.
(2a) C = 1 + Integral 72*x/sqrt(S) dx.
(2b) S = Integral 72*x/sqrt(C) dx.
(2c) C = 1 + Integral S'*sqrt(C/S) dx.
(2d) S = Integral C'*sqrt(S/C) dx.
Exponentials.
(3a) sqrt(C) = exp( Integral 36*x/(C*sqrt(S)) dx ).
(3b) sqrt(S) = 6*x*exp( Integral 36*x/(S*sqrt(C)) - 1/x dx ).
(3c) C - S = exp( Integral 72*x/(C*sqrt(S) + S*sqrt(C)) dx ).
(3d) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
Functional equations.
(4a) C = 1/3 - 36*x^2 + (2/3)*C^(3/2).
(4b) S = 36*x^2 - (2/3)*S^(3/2).
Explicit solutions.
(5a) C(x) = 1 + Sum_{n>=1} 2*(-4)^n*binomial(3*n/2,n)/((3*n-2)*(3*n-4)) * x^n.
(5b) S(x) = 36*x^2 + Sum_{n>=3} 18*(-4)^n*(3*n-3)*binomial(3*n/2-2,n)/((3*n-4)*(3*n-6)) * x^n.
(5c) sqrt(C(x)) = 1 + Sum_{n>=1} -(-4)^n * binomial(3*n/2,n)/(3*n-2) * x^n.
Formulas for terms.
a(n) = -(-4)^n * binomial(3*n/2,n) / (3*n-2) for n>=1, with a(0) = 1.
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