59.1.128 problem 130

Internal problem ID [9300]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 130
Date solved : Monday, January 27, 2025 at 06:01:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.970 (sec). Leaf size: 53

dsolve(x^2*(1+2*x)*diff(y(x),x$2)+x*(5+14*x+3*x^2)*diff(y(x),x)+(4+18*x+12*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {3 x}{2}} \left (\left (2 x +1\right )^{{1}/{4}} \operatorname {HeunC}\left (-\frac {3}{4}, \frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right ) c_{2} +\operatorname {HeunC}\left (-\frac {3}{4}, -\frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right ) c_{1} \right )}{\left (2 x +1\right )^{{1}/{4}} x^{2}} \]

✓ Solution by Mathematica

Time used: 0.337 (sec). Leaf size: 120

DSolve[x^2*(1+2*x)*D[y[x],{x,2}]+x*(5+14*x+3*x^2)*D[y[x],x]+(4+18*x+12*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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