59.1.84 problem 86

Internal problem ID [9256]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 86
Date solved : Monday, January 27, 2025 at 06:00:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.401 (sec). Leaf size: 43

dsolve(12*x^2*(1+x)*diff(y(x),x$2)+x*(11+35*x+3*x^2)*diff(y(x),x)-(1-10*x-5*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {x}{4}} \left (\operatorname {HeunC}\left (\frac {1}{4}, \frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) x^{{7}/{12}} c_{2} +\operatorname {HeunC}\left (\frac {1}{4}, -\frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) c_{1} \right )}{\left (x +1\right )^{{3}/{4}} x^{{1}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.370 (sec). Leaf size: 118

DSolve[12*x^2*(1+x)*D[y[x],{x,2}]+x*(11+35*x+3*x^2)*D[y[x],x]-(1-10*x-5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{24} \left (\frac {3}{K[1]+1}-3+\frac {5}{K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\frac {1}{12} \left (\frac {21}{K[2]+1}+3+\frac {11}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {-3 K[1]^2+5 K[1]+5}{24 K[1]^2+24 K[1]}dK[1]\right )dK[3]+c_1\right ) \]