Internal problem ID [7574]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {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=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 37
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)
\[ y \relax (x ) = c_{1} x^{\frac {1}{4}} \sqrt {x -1}\, \LegendreP \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )+c_{2} x^{\frac {1}{4}} \sqrt {x -1}\, \LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.298 (sec). Leaf size: 550
DSolve[y''[x]== ( -3/(16*x^2) - 2/(9*(x-1)^2) + 3/(16*x*(x-1)) )*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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] \\ \end{align*}