59.1.52 problem 54

Internal problem ID [9224]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 54
Date solved : Monday, January 27, 2025 at 05:52:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 33

dsolve((1+x^2)*diff(y(x),x$2)-8*x*diff(y(x),x)+20*y(x)=0,y(x), singsol=all)
 

\[ y = c_{2} x^{5}+5 c_{1} x^{4}-10 c_{2} x^{3}-10 c_{1} x^{2}+5 c_{2} x +c_{1} \]

✓ Solution by Mathematica

Time used: 0.281 (sec). Leaf size: 77

DSolve[(1+x^2)*D[y[x],{x,2}]-8*x*D[y[x],x]+20*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (x^2+1\right )^2 \exp \left (\int _1^x\frac {K[1]+5 i}{K[1]^2+1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+5 i}{K[1]^2+1}dK[1]\right )dK[2]+c_1\right ) \]