59.1.696 problem 713

Internal problem ID [9868]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 713
Date solved : Monday, January 27, 2025 at 06:15:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 24

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

\[ y = c_{1} \left (x^{2}-1\right )^{{3}/{4}}+c_{2} \left (x^{2}-1\right )^{{1}/{4}} x \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 51

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

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