4.335   ODE No. 1335

\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-1/2\,{\frac { \left ( 3\,x-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{x \left ( x-1 \right ) }}-1/4\,{\frac { \left ( ax+b \right ) y \left ( x \right ) }{x \left ( x-1 \right ) ^{2}}}=0} \]

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

Mathematica: cpu = 0.305039 (sec), leaf count = 893 \[ \left \{\left \{y(x)\to e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} \sqrt [4]{x} c_1 \, _2F_1\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right ),\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4};\frac {1}{2};x\right ) (x-1)^{\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right )+\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4}+\frac {1}{2}\right )}+i e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} x^{3/4} c_2 \, _2F_1\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right )+\frac {1}{2},\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4}+\frac {1}{2};\frac {3}{2};x\right ) (x-1)^{\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right )+\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4}+\frac {1}{2}\right )}\right \}\right \} \]

Maple: cpu = 0.031 (sec), leaf count = 57 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it LegendreP} \left ( { \frac {1}{2}\sqrt {1-4\,a}}-{\frac {1}{2}},\sqrt {-a-b},\sqrt {x} \right ) +{\it \_C2}\,{\it LegendreQ} \left ( {\frac {1}{2}\sqrt {1-4\, a}}-{\frac {1}{2}},\sqrt {-a-b},\sqrt {x} \right ) \right \} \]