59.1.26 problem 26

Internal problem ID [9198]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 26
Date solved : Monday, January 27, 2025 at 05:51:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 58

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

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