Internal problem ID [7163]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 439.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 43
dsolve((4-x^2)*diff(y(x),x$2)+x*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}-4\right )^{\frac {3}{4}} \LegendreP \left (\sqrt {3}-\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right )+c_{2} \left (x^{2}-4\right )^{\frac {3}{4}} \LegendreQ \left (\sqrt {3}-\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 58
DSolve[(4-x^2)*y''[x]+x*y'[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-4\right )^{3/4} \left (c_1 P_{-\frac {1}{2}+\sqrt {3}}^{\frac {3}{2}}\left (\frac {x}{2}\right )+c_2 Q_{-\frac {1}{2}+\sqrt {3}}^{\frac {3}{2}}\left (\frac {x}{2}\right )\right ) \\ \end{align*}