2.1207   ODE No. 1207

\[ x (a x+b) y'(x)+y(x) \left (\text {a1} x^2+\text {b1} x+\text {c1}\right )+x^2 y''(x)=0 \] Mathematica : cpu = 0.0796519 (sec), leaf count = 294


\[\left \{\left \{y(x)\to c_1 U\left (-\frac {-a b+2 \text {b1}-\sqrt {a^2-4 \text {a1}}-\sqrt {a^2-4 \text {a1}} \sqrt {b^2-2 b-4 \text {c1}+1}}{2 \sqrt {a^2-4 \text {a1}}},\sqrt {b^2-2 b-4 \text {c1}+1}+1,\sqrt {a^2-4 \text {a1}} x\right ) \exp \left (\frac {1}{2} \left (-\left (x \left (\sqrt {a^2-4 \text {a1}}+a\right )\right )-\left (-\sqrt {b^2-2 b-4 \text {c1}+1}+b-1\right ) \log (x)\right )\right )+c_2 L_{\frac {-a b+2 \text {b1}-\sqrt {a^2-4 \text {a1}}-\sqrt {a^2-4 \text {a1}} \sqrt {b^2-2 b-4 \text {c1}+1}}{2 \sqrt {a^2-4 \text {a1}}}}^{\sqrt {b^2-2 b-4 \text {c1}+1}}\left (\sqrt {a^2-4 \text {a1}} x\right ) \exp \left (\frac {1}{2} \left (-\left (x \left (\sqrt {a^2-4 \text {a1}}+a\right )\right )-\left (-\sqrt {b^2-2 b-4 \text {c1}+1}+b-1\right ) \log (x)\right )\right )\right \}\right \}\] Maple : cpu = 0.227 (sec), leaf count = 110


\[y \relax (x ) = {\mathrm e}^{-\frac {a x}{2}} x^{-\frac {b}{2}} \left (\WhittakerW \left (-\frac {a b -2 \mathit {b1}}{2 \sqrt {a^{2}-4 \mathit {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}, \sqrt {a^{2}-4 \mathit {a1}}\, x \right ) c_{2}+\WhittakerM \left (-\frac {a b -2 \mathit {b1}}{2 \sqrt {a^{2}-4 \mathit {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}, \sqrt {a^{2}-4 \mathit {a1}}\, x \right ) c_{1}\right )\]