23.4.231 problem 231

Internal problem ID [6533]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 231
Date solved : Friday, October 03, 2025 at 02:09:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 3 x y^{2}-12 x^{2} y y^{\prime }+4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}+8 \left (-x^{3}+1\right ) y y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.212 (sec). Leaf size: 165

ode:=3*x*y(x)^2-12*x^2*y(x)*diff(y(x),x)+4*(-x^3+1)*diff(y(x),x)^2+8*(-x^3+1)*y(x)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ \left (-y^{{3}/{2}} \left (x -1\right ) \left (x^{2}+x +1\right ) \left (\sqrt {10}+1\right ) \operatorname {LegendreQ}\left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+c_1 \sqrt {x}\, \sqrt {-x^{3}+1}\, \sqrt {x^{3}-1}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+\operatorname {LegendreQ}\left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \left (y^{{3}/{2}} \left (x -1\right ) \left (x^{2}+x +1\right ) \left (\sqrt {10}+1\right ) \operatorname {LegendreP}\left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+c_2 \sqrt {x^{3}-1}\, \sqrt {-x^{3}+1}\, \sqrt {x}\right ) &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.504 (sec). Leaf size: 223

ode=3*x*y[x]^2 - 12*x^2*y[x]*D[y[x],x] + 4*(1 - x^3)*D[y[x],x]^2 + 8*(1 - x^3)*y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \exp \left (\int _1^x\frac {3 \left (5 \sqrt [3]{-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{12} \left (17-\sqrt {10}\right ),\frac {1}{12} \left (17+\sqrt {10}\right ),\frac {7}{3},K[1]^3\right ) K[1]-6 c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{12} \left (13-\sqrt {10}\right ),\frac {1}{12} \left (13+\sqrt {10}\right ),\frac {5}{3},K[1]^3\right )\right ) K[1]^2+64 \sqrt [3]{-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{12} \left (5-\sqrt {10}\right ),\frac {1}{12} \left (5+\sqrt {10}\right ),\frac {4}{3},K[1]^3\right )}{96 \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{12} \left (1-\sqrt {10}\right ),\frac {1}{12} \left (1+\sqrt {10}\right ),\frac {2}{3},K[1]^3\right )+\sqrt [3]{-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{12} \left (5-\sqrt {10}\right ),\frac {1}{12} \left (5+\sqrt {10}\right ),\frac {4}{3},K[1]^3\right ) K[1]\right )}dK[1]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-12*x**2*y(x)*Derivative(y(x), x) + 3*x*y(x)**2 + (4 - 4*x**3)*Derivative(y(x), x)**2 + (8 - 8*x**3)*y(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-3*x**2*y(x) + sqrt((-8*x**6*Derivative(y