54.3.389 problem 1406

Internal problem ID [12684]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1406
Date solved : Wednesday, October 01, 2025 at 02:19:45 AM
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*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 44

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

\[ y = \sqrt {x}\, \left (x^{3}-1\right )^{{1}/{4}} \left (c_1 \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+c_2 \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )\right ) \]

✓ Mathematica. Time used: 22.837 (sec). Leaf size: 166

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.212 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(27*x*y(x)/(16*(x**3 - 1)**2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt [4]{x^{3} - 1} \left (C_{1} \sqrt [3]{x^{3}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{12} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{12}, \frac {1}{4} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}\right ) \sqrt [3]{x^{3}}}{x} \]