61.30.18 problem 166

Internal problem ID [12666]
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
Problem number : 166
Date solved : Tuesday, January 28, 2025 at 08:03:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.197 (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)
 

\[ y = \left (a \,x^{2}+b \right )^{-\frac {n}{2}+\frac {1}{4}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.089 (sec). Leaf size: 118

DSolve[(a*x^2+b)*D[y[x],{x,2}]+(2*n+1)*a*x*D[y[x],x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (a x^2+b\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+c_2 Q_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right ) \]