59.1.550 problem 566

Internal problem ID [9722]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 566
Date solved : Monday, January 27, 2025 at 06:13:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 2.119 (sec). Leaf size: 36

dsolve(9*x^2*(1+x)*diff(y(x),x$2)+3*x*(5+11*x-x^2)*diff(y(x),x)+(1+16*x-7*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\frac {c_{1} \operatorname {HeunC}\left (-\frac {1}{3}, -\frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, x +1\right )}{\left (x +1\right )^{{4}/{3}}}+c_{2} \operatorname {HeunC}\left (-\frac {1}{3}, \frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, x +1\right )}{x^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.330 (sec). Leaf size: 120

DSolve[9*x^2*(1+x)*D[y[x],{x,2}]+3*x*(5+11*x-x^2)*D[y[x],x]+(1+16*x-7*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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