Internal problem ID [11012]
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: 178.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 y^{\prime } \left (a x +b \right )+d y=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 100
dsolve((a*x^2+2*b*x+c)*diff(y(x),x$2)+3*(a*x+b)*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} {\left (\sqrt {a \left (a \,x^{2}+2 b x +c \right )}+a x +b \right )}^{\frac {\sqrt {-d +a}}{\sqrt {a}}}}{\sqrt {a \,x^{2}+2 b x +c}}+\frac {c_{2} {\left (\sqrt {a \left (a \,x^{2}+2 b x +c \right )}+a x +b \right )}^{-\frac {\sqrt {-d +a}}{\sqrt {a}}}}{\sqrt {a \,x^{2}+2 b x +c}} \]
✓ Solution by Mathematica
Time used: 0.171 (sec). Leaf size: 152
DSolve[(a*x^2+2*b*x+c)*y''[x]+3*(a*x+b)*y'[x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 P_{\frac {\sqrt {a-d}}{\sqrt {a}}-\frac {1}{2}}^{\frac {1}{2}}\left (\frac {\sqrt {-b^2-a c} (b+a x)}{a \sqrt {c^2-\frac {b^4}{a^2}}}\right )+c_2 Q_{\frac {\sqrt {a-d}}{\sqrt {a}}-\frac {1}{2}}^{\frac {1}{2}}\left (\frac {\sqrt {-b^2-a c} (b+a x)}{a \sqrt {c^2-\frac {b^4}{a^2}}}\right )}{\sqrt [4]{x (a x+2 b)+c}} \]