Answer only with numerics:
SOL = NDSolve[{0.31918 (Piecewise[{{10.268 + 0.0239 x,
x <= 2000}, {58.068, x > 2000}}] -
TAnnulus[x])/(0.00256455 +
1/Piecewise[{{25, x <= 1900}, {100000, x > 1900}}]) -
4.63572 (TAnnulus[x] - TIn[x]) == 3976 D[TAnnulus[x], x],
4.635726619241774` (TAnnulus[x] - TIn[x]) ==
3976 Derivative[1][TIn][x], TAnnulus[0] == 1,
TAnnulus[3000] == TIn[3000]}, {TAnnulus, TIn}, {x, 0, 3000}];
Plot[{TAnnulus[x], TIn[x]} /. SOL // Evaluate, {x, 0, 3000},
PlotLegends -> {"TAnnulus", "TIn"}]
Probably symbolic solution can't be found.