Internal problem ID [10990]
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: 166.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (1+2 n \right ) a x y^{\prime }+c y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 93
dsolve((a*x^2+b)*diff(y(x),x$2)+(2*n+1)*a*x*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (a \,x^{2}+b \right )^{-\frac {n}{2}+\frac {1}{4}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.149 (sec). Leaf size: 118
DSolve[(a*x^2+b)*y''[x]+(2*n+1)*a*x*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (a x^2+b\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+c_2 Q_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right ) \]