59.1.113 problem 115

Internal problem ID [9285]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 115
Date solved : Monday, January 27, 2025 at 06:01:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 46

dsolve(8*x^2*(1+2*x^2)*diff(y(x),x$2)+2*x*(5+34*x^2)*diff(y(x),x)-(1-30*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.359 (sec). Leaf size: 118

DSolve[8*x^2*(1+2*x^2)*D[y[x],{x,2}]+2*x*(5+34*x^2)*D[y[x],x]-(1-30*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\left (\frac {K[1]}{2 K[1]^2+1}+\frac {7}{8 K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\frac {34 K[2]^2+5}{8 K[2]^3+4 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\left (\frac {K[1]}{2 K[1]^2+1}+\frac {7}{8 K[1]}\right )dK[1]\right )dK[3]+c_1\right ) \]