59.1.268 problem 271

Internal problem ID [9440]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 271
Date solved : Monday, January 27, 2025 at 06:02:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 25

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

\[ y = c_{1} x +\arctan \left (\sqrt {x -1}\right ) x c_{2} -\sqrt {x -1}\, c_{2} \]

✓ Solution by Mathematica

Time used: 0.422 (sec). Leaf size: 75

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

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