59.3.5 problem Kovacic 1985 paper. page 23. section 5.2. Example 1
Internal
problem
ID
[10007]
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section
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Problem
number
:
Kovacic
1985
paper.
page
23.
section
5.2.
Example
1
Date
solved
:
Monday, January 27, 2025 at 06:16:41 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*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 30
dsolve(diff(y(x),x$2)= ( -3/(16*x^2) - 2/(9*(x-1)^2) + 3/(16*x*(x-1)) )*y(x),y(x), singsol=all)
\[
y = \sqrt {x -1}\, x^{{1}/{4}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.241 (sec). Leaf size: 550
DSolve[D[y[x],{x,2}]== ( -3/(16*x^2) - 2/(9*(x-1)^2) + 3/(16*x*(x-1)) )*y[x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_1 \exp \left (\int _1^x\text {Root}\left [2048 K[1]^4-3484 K[1]^3+2313 K[1]^2-702 K[1]+\left (20736 K[1]^8-82944 K[1]^7+124416 K[1]^6-82944 K[1]^5+20736 K[1]^4\right ) \text {$\#$1}^4+\left (-48384 K[1]^7+165888 K[1]^6-207360 K[1]^5+110592 K[1]^4-20736 K[1]^3\right ) \text {$\#$1}^3+\left (41472 K[1]^6-118368 K[1]^5+120096 K[1]^4-50976 K[1]^3+7776 K[1]^2\right ) \text {$\#$1}^2+\left (-15360 K[1]^5+34992 K[1]^4-28272 K[1]^3+9936 K[1]^2-1296 K[1]\right ) \text {$\#$1}+81\&,1\right ]dK[1]\right )+c_2 \exp \left (\int _1^x\text {Root}\left [2048 K[1]^4-3484 K[1]^3+2313 K[1]^2-702 K[1]+\left (20736 K[1]^8-82944 K[1]^7+124416 K[1]^6-82944 K[1]^5+20736 K[1]^4\right ) \text {$\#$1}^4+\left (-48384 K[1]^7+165888 K[1]^6-207360 K[1]^5+110592 K[1]^4-20736 K[1]^3\right ) \text {$\#$1}^3+\left (41472 K[1]^6-118368 K[1]^5+120096 K[1]^4-50976 K[1]^3+7776 K[1]^2\right ) \text {$\#$1}^2+\left (-15360 K[1]^5+34992 K[1]^4-28272 K[1]^3+9936 K[1]^2-1296 K[1]\right ) \text {$\#$1}+81\&,1\right ]dK[1]\right ) \int _1^x\exp \left (-2 \int _1^{K[2]}\text {Root}\left [2048 K[1]^4-3484 K[1]^3+2313 K[1]^2-702 K[1]+\left (20736 K[1]^8-82944 K[1]^7+124416 K[1]^6-82944 K[1]^5+20736 K[1]^4\right ) \text {$\#$1}^4+\left (-48384 K[1]^7+165888 K[1]^6-207360 K[1]^5+110592 K[1]^4-20736 K[1]^3\right ) \text {$\#$1}^3+\left (41472 K[1]^6-118368 K[1]^5+120096 K[1]^4-50976 K[1]^3+7776 K[1]^2\right ) \text {$\#$1}^2+\left (-15360 K[1]^5+34992 K[1]^4-28272 K[1]^3+9936 K[1]^2-1296 K[1]\right ) \text {$\#$1}+81\&,1\right ]dK[1]\right )dK[2]
\]