Internal problem ID [11013]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 179.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 482
dsolve((a__2*x^2+b__2*x+c__2)*diff(y(x),x$2)+(b__1*x+c__1)*diff(y(x),x)+c__0*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{3} \operatorname {hypergeom}\left (\left [\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, -\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}\right ], \left [\frac {b_{1} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right )+c_{4} {\left (2 \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 c_{2} a_{2} +b_{2}^{2}\right )}^{\frac {a_{2} \left (a_{2} -\frac {b_{1}}{2}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}} \operatorname {hypergeom}\left (\left [\frac {\frac {a_{2} \left (a_{2} -\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}{2}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}, \frac {\frac {a_{2} \left (a_{2} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}{2}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \left [\frac {2 a_{2} \left (a_{2} -\frac {b_{1}}{4}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 6.771 (sec). Leaf size: 498
DSolve[(a2*x^2+b2*x+c2)*y''[x]+(b1*x+c1)*y'[x]+c0*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\text {a2}-\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {-\text {a2}+\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {\text {b1} \left (\text {b2}+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}\right )-2 \text {a2} \text {c1}}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )-c_2 2^{\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}-\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-1} \exp \left (-\frac {i \pi \left (\text {b1} \left (\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+\text {b2}\right )-2 \text {a2} \text {c1}\right )}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \left (\frac {\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+2 \text {a2} x+\text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )^{-\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}+\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+1} \operatorname {Hypergeometric2F1}\left (\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}-\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},-\frac {\frac {\text {b2} \text {b1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}+\text {a2} \left (-\frac {2 \text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-4\right )}{2 \text {a2}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \]