59.1.249 problem 252

Internal problem ID [9421]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 252
Date solved : Monday, January 27, 2025 at 06:02:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 13

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

\[ y = x \left (\operatorname {Ei}_{1}\left (x \right ) c_{2} +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.443 (sec). Leaf size: 63

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

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