59.1.30 problem 31

Internal problem ID [9202]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 31
Date solved : Monday, January 27, 2025 at 05:52:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 14

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

\[ y = c_{1} x^{2}+{\mathrm e}^{x} c_{2} \]

✓ Solution by Mathematica

Time used: 0.309 (sec). Leaf size: 115

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

\[ y(x)\to \exp \left (\int _1^x\frac {(K[1]-4) K[1]+2}{2 (K[1]-2) K[1]}dK[1]-\frac {1}{2} \int _1^x\left (-\frac {1}{K[2]}-1+\frac {1}{2-K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]^2-4 K[1]+2}{2 (K[1]-2) K[1]}dK[1]\right )dK[3]+c_1\right ) \]