59.1.54 problem 56

Internal problem ID [9226]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 56
Date solved : Monday, January 27, 2025 at 05:52:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 37

dsolve((1+2*x^2)*diff(y(x),x$2)+7*x*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \operatorname {LegendreP}\left (\frac {1}{4}, \frac {3}{4}, i \sqrt {2}\, x \right )+c_{2} \operatorname {LegendreQ}\left (\frac {1}{4}, \frac {3}{4}, i \sqrt {2}\, x \right )}{\left (2 x^{2}+1\right )^{{3}/{8}}} \]

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 66

DSolve[(1+2*x^2)*D[y[x],{x,2}]+7*x*D[y[x],x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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