54.7.160 problem 1777 (book 6.186)

Internal problem ID [13009]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1777 (book 6.186)
Date solved : Friday, October 03, 2025 at 03:58:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.234 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 53.454 (sec). Leaf size: 540

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 {\exp \left (-2 \int _1^{K[3]}\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 (8 \left (K[3]^3-1\right )+\exp \left (2 \int _1^{K[3]}\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 ) c_1 \left (2 K[3]^2+2 K[3]+\sqrt {2 K[3]+i \sqrt {3}+1} \sqrt {6 K[3]-3 i \sqrt {3}+3}+2\right )+\exp \left (2 \int _1^{K[3]}\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 (2 K[3]^2+2 K[3]+\sqrt {2 K[3]+i \sqrt {3}+1} \sqrt {6 K[3]-3 i \sqrt {3}+3}+2\right ) \int _1^{K[3]}\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]\right )}{4 \left (K[3]^3-1\right ) \left (c_1+\int _1^{K[3]}\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]\right )}dK[3]\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(x), (x, 2)) + 6*x**4*y(x) - 16*x**3*Derivative(y(x), (x, 2)) + 3*x*y(x) + 8*Derivative(y(x), (x, 2)))*y(x)))/(2*(x**3 - 1)) cannot be solved by the factorable group method