Internal problem ID [11011]
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: 177.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+y^{\prime } \left (a x +b \right )+d y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 77
dsolve((a*x^2+2*b*x+c)*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\frac {a x +b}{\sqrt {a}}+\sqrt {a \,x^{2}+2 b x +c}\right )^{\frac {i \sqrt {d}}{\sqrt {a}}}+c_{2} \left (\frac {a x +b}{\sqrt {a}}+\sqrt {a \,x^{2}+2 b x +c}\right )^{-\frac {i \sqrt {d}}{\sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.404 (sec). Leaf size: 93
DSolve[(a*x^2+2*b*x+c)*y''[x]+(a*x+b)*y'[x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cos \left (\frac {\sqrt {d} \log \left (-\sqrt {a} \sqrt {a x^2+2 b x+c}+a x+b\right )}{\sqrt {a}}\right )-c_2 \sin \left (\frac {\sqrt {d} \log \left (-\sqrt {a} \sqrt {a x^2+2 b x+c}+a x+b\right )}{\sqrt {a}}\right ) \]