59.1.242 problem 245

Internal problem ID [9414]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 245
Date solved : Monday, January 27, 2025 at 06:02:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 0.282 (sec). Leaf size: 80

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

\begin{align*} y(x)\to \exp \left (\int _1^x\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}\left (1-\frac {1}{K[1]}\right )dK[1]\right ) c_1dK[2]+c_2\right ) \\ y(x)\to c_2 \exp \left (\int _1^x\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \\ \end{align*}