59.1.105 problem 107

Internal problem ID [9277]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 107
Date solved : Monday, January 27, 2025 at 06:00:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 35

dsolve(2*x^2*(2+x^2)*diff(y(x),x$2)+x*(4+7*x^2)*diff(y(x),x)-(1-3*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.243 (sec). Leaf size: 95

DSolve[2*x^2*(2+x^2)*D[y[x],{x,2}]+x*(4+7*x^2)*D[y[x],x]-(1-3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (c_2 \sqrt [8]{x^2+2} \operatorname {Gamma}\left (\frac {3}{4}\right ) Q_{-\frac {1}{4}}^{\frac {1}{4}}\left (\frac {i x}{\sqrt {2}}\right )+2^{3/8} c_1\right ) \exp \left (\int _1^x-\frac {3 K[1]^2+4}{4 K[1]^3+8 K[1]}dK[1]\right )}{\sqrt [8]{x^2+2} \operatorname {Gamma}\left (\frac {3}{4}\right )} \]