54.7.206 problem 1831 (book 6.240)

Internal problem ID [13055]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1831 (book 6.240)
Date solved : Friday, October 03, 2025 at 03:59:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y^{\prime \prime } y-36 x {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.434 (sec). Leaf size: 232

ode:=x^2*(2-9*x)*diff(diff(y(x),x),x)^2-6*x*(1-6*x)*diff(y(x),x)*diff(diff(y(x),x),x)+6*diff(diff(y(x),x),x)*y(x)-36*x*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {9 c_{1} \sqrt {\frac {-1+5 x +\sqrt {9 x^{2}-2 x}}{\sqrt {9 x^{2}-2 x}\, \sqrt {-\frac {\left (4 x -1\right )^{2}}{x \left (9 x -2\right )}}}}\, \sqrt {4 x -1}\, x}{\left (-1+9 x +3 \sqrt {9 x^{2}-2 x}\right ) \sqrt {27 x -3+9 \sqrt {9 x^{2}-2 x}}} \\ y &= \frac {c_{1} \left (-1+9 x +3 \sqrt {9 x^{2}-2 x}\right ) \sqrt {27 x -3+9 \sqrt {9 x^{2}-2 x}}\, \sqrt {4 x -1}\, x}{9 \sqrt {\frac {-1+5 x +\sqrt {9 x^{2}-2 x}}{\sqrt {9 x^{2}-2 x}\, \sqrt {-\frac {\left (4 x -1\right )^{2}}{x \left (9 x -2\right )}}}}} \\ y &= 0 \\ y &= c_{2} x^{3}+c_{1} x +\frac {c_{1}^{2}}{c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 29

ode=-36*x*D[y[x],x]^2 + 6*y[x]*D[y[x],{x,2}] - 6*(1 - 6*x)*x*D[y[x],x]*D[y[x],{x,2}] + (2 - 9*x)*x^2*D[y[x],{x,2}]^2 == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1{}^2 x^3}{c_2}+c_1 x+c_2\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2 - 9*x)*Derivative(y(x), (x, 2))**2 - 6*x*(1 - 6*x)*Derivative(y(x), x)*Derivative(y(x), (x, 2)) - 36*x*Derivative(y(x), x)**2 + 6*y(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(6*x - 1)*Derivative(y(x), (x, 2)) + sqrt(x*(-4*x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), (x, 2)) + 24*y(x))*Derivative(y(x), (x, 2))))/(12*x) cannot be solved by the factorable group method