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It can be solved numerically in Mathematica using NDSolve , although a working precision of 40 decimal digits was used to obtain the results shown.

Intrinsically defined curves
With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 )
NDSolve[{x'[s] Cos[ θ [s]], y'[s] Sin[ θ [s]], θ '[s] f[s], x[0] y[0] θ [0] 0}, {x, y, θ }, {s, 0, s max }]
For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve :
f[s] = 1: {Sin[ θ ], Cos[ θ ]}
f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]}
f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]}
f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]}
f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]}
f[s] = s n : result involves Gamma[1/n, ± θ n/n ]
f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

Three-body problem
The two-body problem was analyzed by Johannes Kepler in 1609 and solved by Isaac Newton in 1687. … (It is now possible to get most results just by direct numerical computation using for example NDSolve .) … And from these results the conclusion was drawn that the three-body problem could not be solved in terms of algebraic formulas and integrals.

Implementation [of my PDEs]
All the numerical solutions shown were found using the NDSolve function built into Mathematica.

Roots of polynomials can thus almost always be found with NSolve in about Log[n] m[n] operations. If one evaluates NIntegrate or NDSolve by effectively fitting functions to order s polynomials the difficulty of getting results with n -digit precision typically increases like 2 n/s .