59.1.85 problem 87
Internal
problem
ID
[9257]
Book
:
Collection
of
Kovacic
problems
Section
:
section
1
Problem
number
:
87
Date
solved
:
Monday, January 27, 2025 at 06:00:37 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 2.530 (sec). Leaf size: 162
dsolve(x^2*(5+x+10*x^2)*diff(y(x),x$2)+x*(4+3*x+48*x^2)*diff(y(x),x)+(x+36*x^2)*y(x)=0,y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{-\frac {\sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}} \left (i \sqrt {199}-20 x -1\right )^{\frac {i \sqrt {199}}{1990}} \left (i \sqrt {199}+20 x +1\right )^{-\frac {i \sqrt {199}}{1990}} \left (\operatorname {HeunG}\left (\frac {\sqrt {199}+i}{i-\sqrt {199}}, 0, 0, \frac {1}{5}, \frac {6}{5}, -\frac {i \sqrt {199}}{995}, -\frac {20 x}{1+i \sqrt {199}}\right ) x^{{1}/{5}} c_{2} +\operatorname {HeunG}\left (\frac {\sqrt {199}+i}{i-\sqrt {199}}, \frac {15721-179 i \sqrt {199}}{194275 i \sqrt {199}+641775}, -\frac {1}{5}, 0, \frac {4}{5}, -\frac {i \sqrt {199}}{995}, -\frac {20 x}{1+i \sqrt {199}}\right ) c_{1} \right )}{10 x^{2}+x +5}
\]
✓ Solution by Mathematica
Time used: 1.139 (sec). Leaf size: 132
DSolve[x^2*(5+x+10*x^2)*D[y[x],{x,2}]+x*(4+3*x+48*x^2)*D[y[x],x]+(x+36*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \exp \left (\int _1^x\left (\frac {3}{5 K[1]}-\frac {1}{10 \left (10 K[1]^2+K[1]+5\right )}\right )dK[1]-\frac {1}{2} \int _1^x\frac {48 K[2]^2+3 K[2]+4}{10 K[2]^3+K[2]^2+5 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\left (\frac {3}{5 K[1]}-\frac {1}{10 \left (10 K[1]^2+K[1]+5\right )}\right )dK[1]\right )dK[3]+c_1\right )
\]