59.1.86 problem 88

Internal problem ID [9258]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 88
Date solved : Monday, January 27, 2025 at 06:00:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.383 (sec). Leaf size: 38

dsolve(18*x^2*(1+x)*diff(y(x),x$2)+3*x*(5+11*x+x^2)*diff(y(x),x)-(1-2*x-5*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {x}{6}} \left (\sqrt {x}\, \operatorname {HeunC}\left (\frac {1}{6}, \frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right ) c_{2} +\operatorname {HeunC}\left (\frac {1}{6}, -\frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right ) c_{1} \right )}{x^{{1}/{6}}} \]

✓ Solution by Mathematica

Time used: 0.368 (sec). Leaf size: 118

DSolve[18*x^2*(1+x)*D[y[x],{x,2}]+3*x*(5+11*x+x^2)*D[y[x],x]-(1-2*x-5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{12} \left (\frac {7}{K[1]+1}-1+\frac {3}{K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\frac {1}{6} \left (\frac {5}{K[2]+1}+1+\frac {5}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {-K[1]^2+9 K[1]+3}{12 K[1]^2+12 K[1]}dK[1]\right )dK[3]+c_1\right ) \]