Internal problem ID [10919]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 85.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 165
dsolve(x*diff(y(x),x$2)+(a*x^2+b*x+2)*diff(y(x),x)+(c*x^2+d*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \operatorname {hypergeom}\left (\left [\frac {2 a^{3}-d \,a^{2}+a b c -c^{2}}{2 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) {\mathrm e}^{-\frac {x \left (a^{2} x +2 a b -2 c \right )}{2 a}}}{x}+\frac {c_{2} \left (a^{2} x +a b -2 c \right ) \operatorname {hypergeom}\left (\left [\frac {3 a^{3}-d \,a^{2}+a b c -c^{2}}{2 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) {\mathrm e}^{-\frac {x \left (a^{2} x +2 a b -2 c \right )}{2 a}}}{x} \]
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 134
DSolve[x*y''[x]+(a*x^2+b*x+2)*y'[x]+(c*x^2+d*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {1}{2} x \left (-\frac {2 c}{a}+a x+2 b\right )} \left (c_2 \operatorname {Hypergeometric1F1}\left (-\frac {-2 a^3+d a^2-b c a+c^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+b a-2 c\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-2 a^3+d a^2-b c a+c^2}{a^3},\frac {x a^2+b a-2 c}{\sqrt {2} a^{3/2}}\right )\right )}{x} \]