23.4.297 problem 300

Internal problem ID [6599]
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 : 300
Date solved : Friday, October 03, 2025 at 02:09:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.351 (sec). Leaf size: 232

ode:=6*y(x)*diff(diff(y(x),x),x)-6*(1-6*x)*x*diff(y(x),x)*diff(diff(y(x),x),x)+(2-9*x)*x^2*diff(diff(y(x),x),x)^2 = 36*x*diff(y(x),x)^2; 
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 )}}}}\, x \sqrt {4 x -1}}{\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}}\, x \sqrt {4 x -1}}{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: 0.024 (sec). Leaf size: 29

ode=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 == 36*x*D[y[x],x]^2; 
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 - x*(6 - 36*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)) + sq