59.1.502 problem 518
Internal
problem
ID
[9674]
Book
:
Collection
of
Kovacic
problems
Section
:
section
1
Problem
number
:
518
Date
solved
:
Monday, January 27, 2025 at 06:12:55 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.654 (sec). Leaf size: 231
dsolve(2*x^2*(1+x+x^2)*diff(y(x),x$2)+x*(9+11*x+11*x^2)*diff(y(x),x)+(6+10*x+7*x^2)*y(x)=0,y(x), singsol=all)
\[
y = \frac {\left (2 x +i \sqrt {3}+1\right )^{\frac {5 \sqrt {3}+3 i}{6 \sqrt {3}+6 i}} \left (-2 x +i \sqrt {3}-1\right )^{\frac {64 i \sqrt {3}+2368}{\left (\sqrt {3}+i\right )^{3} \left (i-\sqrt {3}\right )^{4} \left (13 \sqrt {3}+9 i\right )}} {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}} \left (\operatorname {HeunG}\left (\frac {\sqrt {3}+i}{i-\sqrt {3}}, 0, 0, \frac {5}{2}, \frac {1}{2}, \frac {5 \sqrt {3}+3 i}{3 \sqrt {3}+3 i}, -\frac {2 x}{1+i \sqrt {3}}\right ) c_{1} \sqrt {x}+\operatorname {HeunG}\left (\frac {\sqrt {3}+i}{i-\sqrt {3}}, -\frac {64}{\left (i \sqrt {3}-1\right )^{3} \left (i-\sqrt {3}\right )^{4}}, \frac {1}{2}, 3, \frac {3}{2}, \frac {5 \sqrt {3}+3 i}{3 \sqrt {3}+3 i}, -\frac {2 x}{1+i \sqrt {3}}\right ) c_{2} x \right )}{x^{{5}/{2}} \left (x^{2}+x +1\right )^{{1}/{4}}}
\]
✓ Solution by Mathematica
Time used: 0.367 (sec). Leaf size: 135
DSolve[2*x^2*(1+x+x^2)*D[y[x],{x,2}]+x*(9+11*x+11*x^2)*D[y[x],x]+(6+10*x+7*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \exp \left (\int _1^x\frac {K[1] (7 K[1]+3)+1}{4 K[1] \left (K[1]^2+K[1]+1\right )}dK[1]-\frac {1}{2} \int _1^x\left (\frac {K[2]+1}{K[2]^2+K[2]+1}+\frac {9}{2 K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {7 K[1]^2+3 K[1]+1}{4 K[1] \left (K[1]^2+K[1]+1\right )}dK[1]\right )dK[3]+c_1\right )
\]