\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac {by \left ( x \right ) }{{x}^{2} \left ( x-a \right ) ^{2}}}+c=0} \]
Mathematica: cpu = 0.626080 (sec), leaf count = 589 \[ \left \{\left \{y(x)\to -\frac {2 c x^2 (a-x) \left (1-\frac {x}{a}\right )^{-\frac {1}{2} \sqrt {\frac {a^2-4 b}{a^2}}} \left (\sqrt {\frac {a^2-4 b}{a^2}} \left (1-\frac {x}{a}\right )^{\sqrt {\frac {a^2-4 b}{a^2}}} \, _2F_1\left (\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {3}{2};\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {5}{2};\frac {x}{a}\right )-3 \left (1-\frac {x}{a}\right )^{\sqrt {\frac {a^2-4 b}{a^2}}} \, _2F_1\left (\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {3}{2};\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {5}{2};\frac {x}{a}\right )+\sqrt {\frac {a^2-4 b}{a^2}} \, _2F_1\left (-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {3}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}};\frac {5}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}};\frac {x}{a}\right )+3 \, _2F_1\left (-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {3}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}};\frac {5}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}};\frac {x}{a}\right )\right )}{a \left (\sqrt {\frac {a^2-4 b}{a^2}}-3\right ) \left (\sqrt {\frac {a^2-4 b}{a^2}}+3\right ) \sqrt {\frac {a^2-4 b}{a^2}} \sqrt {\frac {a-x}{a}}}+\frac {c_2 (x-a)^{\frac {1}{2} \sqrt {\frac {a^2-4 b}{a^2}}+\frac {1}{2}} x^{\frac {1}{2}-\frac {1}{2} \sqrt {\frac {a^2-4 b}{a^2}}}}{a \sqrt {\frac {a^2-4 b}{a^2}}}+c_1 (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} x^{\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {1}{2}}\right \}\right \} \]
Maple: cpu = 0.109 (sec), leaf count = 219 \[ \left \{ y \left ( x \right ) =\sqrt {x \left ( a-x \right ) } \left ( { \frac {a-x}{x}} \right ) ^{{\frac {1}{2\,a}\sqrt {{a}^{2}-4\,b}}}{\it \_C2}+\sqrt {x \left ( a-x \right ) } \left ( {\frac {x}{a-x}} \right ) ^{ {\frac {1}{2\,a}\sqrt {{a}^{2}-4\,b}}}{\it \_C1}+{c\sqrt {x \left ( a-x \right ) } \left ( \left ( {\frac {x}{a-x}} \right ) ^{{\frac {1}{2\,a} \sqrt {{a}^{2}-4\,b}}}\int \!\sqrt {x \left ( a-x \right ) } \left ( { \frac {x}{a-x}} \right ) ^{-{\frac {1}{2\,a}\sqrt {{a}^{2}-4\,b}}} \,{\rm d}x- \left ( {\frac {a-x}{x}} \right ) ^{{\frac {1}{2\,a}\sqrt {{ a}^{2}-4\,b}}}\int \!\sqrt {x \left ( a-x \right ) } \left ( {\frac {a-x }{x}} \right ) ^{-{\frac {1}{2\,a}\sqrt {{a}^{2}-4\,b}}}\,{\rm d}x \right ) {\frac {1}{\sqrt {{a}^{2}-4\,b}}}} \right \} \]