g[x_] := 1/(11664 a^4)
E^(-(((5 + 3 Sqrt[30]) a)/(35 x))) (36 (-5 c E^(((5 + 3 Sqrt[30]) a)/(35 x)) (324 a^2 - 630 a x - 7595 x^2) + 324 a^4 (E^((6 Sqrt[6/5] a)/(7 x)) (18 a^2 - 21 Sqrt[30] a x + 245 x^2) C1 + (18 a^2 + 21 Sqrt[30] a x + 245 x^2) C2)) + 245 c (E^((6 Sqrt[6/5] a)/(7 x)) (18 Sqrt[30] a^2 - 630 a x + 245 Sqrt[30] x^2) ExpIntegralEi[((5 - 3 Sqrt[30]) a)/(35 x)] - (18 Sqrt[30] a^2 + 630 a x + 245 Sqrt[30] x^2) ExpIntegralEi[((5 + 3 Sqrt[30]) a)/(35 x)]));

C1 = -((2790 c + 5832 a^4 C2 - 245 Sqrt[30] c ArcCoth[3Sqrt[6/5]])/(5832 a^4));

C2 =-(5 c (44100 (434 + 67 a) E^(1/35 (5 + 3 Sqrt[30]) a) + 2 E^(6/7 Sqrt[6/5]a) (558 (-17150 + a (245 (-5 + 6 Sqrt[30]) + 3 a (35 (-18 + Sqrt[30]) + 6 (-5 + 3 Sqrt[30]) a))) + 245 (3430 Sqrt[30] +  a (245 (-36 + Sqrt[30]) + 18 a (-35 + 21 Sqrt[30] - 18 a + Sqrt[30] a))) ArcCoth[3 Sqrt[6/5]]) + 245 (3430 Sqrt[30] + a (245 (-36 + Sqrt[30]) + 18 a (7 (-5 + 3 Sqrt[30]) + (-18 + Sqrt[30]) a))) E^( 6/7 Sqrt[6/5] a) ExpIntegralEi[1/35 (5 - 3 Sqrt[30]) a] - 245 (3430 Sqrt[30] + a (245 (36 + Sqrt[30]) + 18 a (35 + 21 Sqrt[30] + 18 a +  Sqrt[30] a))) ExpIntegralEi[1/35 (5 + 3 Sqrt[30]) a]))/(11664 a^4 (17150 + 1225 a + 1470 Sqrt[30] a + 1890 a^2 + 105 Sqrt[30] a^2 + 90 a^3 + 54 Sqrt[30]a^3 + (-17150 + a (245 (-5 + 6 Sqrt[30]) + 3 a (35 (-18 + Sqrt[30]) + 6 (-5 + 3 Sqrt[30]) a))) E^(6/7 Sqrt[6/5] a)));

Simplify[DSolve[yy'[x] == a a20/5 g[x]/x^2 + c/x^2, yy[x], x ]]