59.1.58 problem 60

Internal problem ID [9230]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 60
Date solved : Monday, January 27, 2025 at 05:52:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 37

dsolve((1+2*x^2)*diff(y(x),x$2)-9*x*diff(y(x),x)-6*y(x)=0,y(x), singsol=all)
 

\[ y = \left (\operatorname {LegendreP}\left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right ) c_{1} +\operatorname {LegendreQ}\left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right ) c_{2} \right ) \left (2 x^{2}+1\right )^{{13}/{8}} \]

✓ Solution by Mathematica

Time used: 0.236 (sec). Leaf size: 71

DSolve[(1+2*x^2)*D[y[x],{x,2}]-9*x*D[y[x],x]-6*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \left (2 x^2+1\right )^{13/8} Q_{\frac {11}{4}}^{\frac {13}{4}}\left (i \sqrt {2} x\right )+\frac {64 \sqrt [4]{2} c_1 \left (3 x^6+5 x^4+3 x^2+1\right )}{3 \operatorname {Gamma}\left (-\frac {9}{4}\right )} \]