59.1.121 problem 123

Internal problem ID [9293]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 123
Date solved : Monday, January 27, 2025 at 06:01:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 45

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

\[ y = \frac {x^{2} \left (\operatorname {Ei}_{1}\left (-\frac {4 x}{x -1}\right ) {\mathrm e}^{-\frac {4 x}{x -1}} c_{2} +{\mathrm e}^{-\frac {4}{x -1}} c_{1} \right )}{x -1} \]

✓ Solution by Mathematica

Time used: 0.305 (sec). Leaf size: 116

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

\[ y(x)\to \exp \left (\int _1^x\frac {2 K[1]^2+K[1]+1}{2 (K[1]-1)^2 K[1]}dK[1]-\frac {1}{2} \int _1^x-\frac {K[2]+3}{(K[2]-1)^2 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2 K[1]^2+K[1]+1}{2 (K[1]-1)^2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]