59.1.83 problem 85

Internal problem ID [9255]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 85
Date solved : Monday, January 27, 2025 at 06:00:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.280 (sec). Leaf size: 38

dsolve(3*x^2*diff(y(x),x$2)+2*x*(1+x-2*x^2)*diff(y(x),x)+(2*x-8*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} x^{{1}/{3}} {\mathrm e}^{\frac {2 \left (x -1\right ) x}{3}}+c_{2} \operatorname {HeunB}\left (-\frac {1}{3}, \frac {\sqrt {6}}{3}, -\frac {7}{3}, \frac {4 \sqrt {6}}{9}, -\frac {\sqrt {6}\, x}{3}\right ) \]

✓ Solution by Mathematica

Time used: 0.943 (sec). Leaf size: 53

DSolve[3*x^2*D[y[x],{x,2}]+2*x*(1+x-2*x^2)*D[y[x],x]+(2*x-8*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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