\[ \boxed { x{\frac {{\rm d}^{5}}{{\rm d}{x}^{5}}}y \left ( x \right ) -5\,m{\frac {{\rm d}^{4}}{{\rm d}{x}^{4}}}y \left ( x \right ) +axy \left ( x \right ) =0} \]
Mathematica: cpu = 2.837860 (sec), leaf count = 216 \[ \left \{\left \{y(x)\to c_5 5^{-5 m-4} a^{\frac {1}{5} (5 m+4)} x^{5 m+4} \, _0F_4\left (;m+\frac {6}{5},m+\frac {7}{5},m+\frac {8}{5},m+\frac {9}{5};-\frac {a x^5}{3125}\right )+\frac {1}{125} a^{3/5} c_4 x^3 \, _0F_4\left (;\frac {6}{5},\frac {7}{5},\frac {8}{5},\frac {4}{5}-m;-\frac {a x^5}{3125}\right )+\frac {1}{25} a^{2/5} c_3 x^2 \, _0F_4\left (;\frac {4}{5},\frac {6}{5},\frac {7}{5},\frac {3}{5}-m;-\frac {a x^5}{3125}\right )+\frac {1}{5} \sqrt [5]{a} c_2 x \, _0F_4\left (;\frac {3}{5},\frac {4}{5},\frac {6}{5},\frac {2}{5}-m;-\frac {a x^5}{3125}\right )+c_1 \, _0F_4\left (;\frac {2}{5},\frac {3}{5},\frac {4}{5},\frac {1}{5}-m;-\frac {a x^5}{3125}\right )\right \}\right \} \]
Maple: cpu = 0.124 (sec), leaf count = 118 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, {\mbox {$_0$F$_4$}(\ ;\,{\frac {2}{5}},{\frac {3}{5}},{\frac {4}{5}},{\frac {1}{5}}-m;\,-{\frac {{x}^{5}a}{3125}})} +{\it \_C2}\,x {\mbox {$_0$F$_4$}(\ ;\,{\frac {3}{5}},{\frac {4}{5}},{\frac {6}{5}},{\frac {2}{5}}-m;\,-{\frac {{x}^{5}a}{3125}})} +{\it \_C3}\,{x}^{2} {\mbox {$_0$F$_4$}(\ ;\,{\frac {4}{5}},{\frac {6}{5}},{\frac {7}{5}},{\frac {3}{5}}-m;\,-{\frac {{x}^{5}a}{3125}})} +{\it \_C4}\,{x}^{3} {\mbox {$_0$F$_4$}(\ ;\,{\frac {6}{5}},{\frac {7}{5}},{\frac {8}{5}},{\frac {4}{5}}-m;\,-{\frac {{x}^{5}a}{3125}})} +{\it \_C5}\,{x}^{4+5\,m} {\mbox {$_0$F$_4$}(\ ;\,{\frac {9}{5}}+m,{\frac {8}{5}}+m,{\frac {7}{5}}+m,{\frac {6}{5}}+m;\,-{\frac {{x}^{5}a}{3125}})} \right \} \]