Internal problem ID [11010]
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: 176.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 315
dsolve((a*x^2+b*x+c)*diff(y(x),x$2)+(k*x+d)*diff(y(x),x)-k*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (k x +d \right )+c_{2} {\left (2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, x \,a^{2}+\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, b a -4 a c +b^{2}\right )}^{\frac {a \left (a -\frac {k}{2}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+a d -\frac {b k}{2}}{\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}} \operatorname {hypergeom}\left (\left [-\frac {k \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a -2 a d +b k}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}, \frac {a \left (a +\frac {k}{2}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+a d -\frac {b k}{2}}{\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \left [\frac {4 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}-k \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a +2 a d -b k}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+4 a c -b^{2}}{8 a c -2 b^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 4.256 (sec). Leaf size: 107
DSolve[(a*x^2+b*x+c)*y''[x]+(k*x+d)*y'[x]-k*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(d+k x) \left (c_2 \int _1^x\frac {\exp \left (\frac {(b k-2 a d) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) (c+K[1] (b+a K[1]))^{-\frac {k}{2 a}}}{(d+k K[1])^2}dK[1]+c_1\right )}{d} \]