60.3.400 problem 1406

Internal problem ID [11410]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1406
Date solved : Monday, January 27, 2025 at 11:19:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {27 x y}{16 \left (x^{3}-1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 44

dsolve(diff(diff(y(x),x),x) = -27/16*x/(x^3-1)^2*y(x),y(x), singsol=all)
 

\[ y = \sqrt {x}\, \left (x^{3}-1\right )^{{1}/{4}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )\right ) \]

✓ Solution by Mathematica

Time used: 23.357 (sec). Leaf size: 166

DSolve[D[y[x],{x,2}] == (-27*x*y[x])/(16*(-1 + x^3)^2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right )dK[2]+c_1\right ) \]