59.3.5 problem Kovacic 1985 paper. page 23. section 5.2. Example 1
Internal
problem
ID
[10007]
Book
:
Collection
of
Kovacic
problems
Section
:
section
3.
Problems
from
Kovacic
related
papers
Problem
number
:
Kovacic
1985
paper.
page
23.
section
5.2.
Example
1
Date
solved
:
Sunday, March 30, 2025 at 02:51:28 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }&=\left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (x -1\right )^{2}}+\frac {3}{16 x \left (x -1\right )}\right ) y \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 30
ode:=diff(diff(y(x),x),x) = (-3/16/x^2-2/9/(x-1)^2+3/16/x/(x-1))*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \sqrt {x -1}\, x^{{1}/{4}} \left (c_1 \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )+c_2 \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )\right )
\]
✓ Mathematica. Time used: 0.241 (sec). Leaf size: 550
ode=D[y[x],{x,2}]== ( -3/(16*x^2) - 2/(9*(x-1)^2) + 3/(16*x*(x-1)) )*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 \exp \left (\int _1^x\text {Root}\left [2048 K[1]^4-3484 K[1]^3+2313 K[1]^2-702 K[1]+\left (20736 K[1]^8-82944 K[1]^7+124416 K[1]^6-82944 K[1]^5+20736 K[1]^4\right ) \text {$\#$1}^4+\left (-48384 K[1]^7+165888 K[1]^6-207360 K[1]^5+110592 K[1]^4-20736 K[1]^3\right ) \text {$\#$1}^3+\left (41472 K[1]^6-118368 K[1]^5+120096 K[1]^4-50976 K[1]^3+7776 K[1]^2\right ) \text {$\#$1}^2+\left (-15360 K[1]^5+34992 K[1]^4-28272 K[1]^3+9936 K[1]^2-1296 K[1]\right ) \text {$\#$1}+81\&,1\right ]dK[1]\right )+c_2 \exp \left (\int _1^x\text {Root}\left [2048 K[1]^4-3484 K[1]^3+2313 K[1]^2-702 K[1]+\left (20736 K[1]^8-82944 K[1]^7+124416 K[1]^6-82944 K[1]^5+20736 K[1]^4\right ) \text {$\#$1}^4+\left (-48384 K[1]^7+165888 K[1]^6-207360 K[1]^5+110592 K[1]^4-20736 K[1]^3\right ) \text {$\#$1}^3+\left (41472 K[1]^6-118368 K[1]^5+120096 K[1]^4-50976 K[1]^3+7776 K[1]^2\right ) \text {$\#$1}^2+\left (-15360 K[1]^5+34992 K[1]^4-28272 K[1]^3+9936 K[1]^2-1296 K[1]\right ) \text {$\#$1}+81\&,1\right ]dK[1]\right ) \int _1^x\exp \left (-2 \int _1^{K[2]}\text {Root}\left [2048 K[1]^4-3484 K[1]^3+2313 K[1]^2-702 K[1]+\left (20736 K[1]^8-82944 K[1]^7+124416 K[1]^6-82944 K[1]^5+20736 K[1]^4\right ) \text {$\#$1}^4+\left (-48384 K[1]^7+165888 K[1]^6-207360 K[1]^5+110592 K[1]^4-20736 K[1]^3\right ) \text {$\#$1}^3+\left (41472 K[1]^6-118368 K[1]^5+120096 K[1]^4-50976 K[1]^3+7776 K[1]^2\right ) \text {$\#$1}^2+\left (-15360 K[1]^5+34992 K[1]^4-28272 K[1]^3+9936 K[1]^2-1296 K[1]\right ) \text {$\#$1}+81\&,1\right ]dK[1]\right )dK[2]
\]
✓ Sympy. Time used: 0.348 (sec). Leaf size: 41
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(-2/(9*(x - 1)**2) + 3/(16*x*(x - 1)) - 3/(16*x**2))*y(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \sqrt [4]{x} \sqrt [3]{x - 1} \left (C_{1} \sqrt {x} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{12}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {x} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{12}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x} \right )}\right )
\]