59.1.695 problem 712

Internal problem ID [9867]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 712
Date solved : Monday, January 27, 2025 at 06:15:13 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 27

dsolve(x*(1-x)*diff(y(x),x$2)+1/3*(1-2*x)*diff(y(x),x)+20/9*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \left (6 x -5\right ) x^{{2}/{3}}+c_{2} \left (6 x -1\right ) \left (x -1\right )^{{2}/{3}} \]

✓ Solution by Mathematica

Time used: 0.494 (sec). Leaf size: 93

DSolve[x*(1-x)*D[y[x],{x,2}]+1/3*(1-2*x)*D[y[x],x]+20/9*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (3 c_2 (x-1) \operatorname {Gamma}\left (\frac {4}{3}\right ) Q_1^{\frac {2}{3}}(2 x-1)+c_1 (5-6 x) (1-x)^{2/3} \sqrt [3]{x}\right ) \exp \left (\int _1^x\frac {1-2 K[1]}{3 K[1]-3 K[1]^2}dK[1]\right )}{3 (x-1) \operatorname {Gamma}\left (\frac {4}{3}\right )} \]