59.1.55 problem 57

Internal problem ID [9227]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 57
Date solved : Monday, January 27, 2025 at 05:52:18 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 39

dsolve((1-x^2)*diff(y(x),x$2)-5*x*diff(y(x),x)-4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 47

DSolve[(1-x^2)*D[y[x],{x,2}]-5*x*D[y[x],x]-4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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