23.4.304 problem 307

Internal problem ID [6606]
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 : 307
Date solved : Tuesday, September 30, 2025 at 03:27:27 PM
CAS classification : unknown

\begin{align*} 32 y^{\prime \prime } \left (-y^{\prime }+x y^{\prime \prime }\right )^{3}+\left (-{y^{\prime }}^{2}+2 y y^{\prime \prime }\right )^{3}&=0 \end{align*}

✗ Maple

ode:=32*diff(diff(y(x),x),x)*(-diff(y(x),x)+x*diff(diff(y(x),x),x))^3+(-diff(y(x),x)^2+2*y(x)*diff(diff(y(x),x),x))^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 41.8 (sec). Leaf size: 142

ode=32*D[y[x],{x,2}]*(-D[y[x],x] + x*D[y[x],{x,2}])^3 + (-D[y[x],x]^2 + 2*y[x]*D[y[x],{x,2}])^3 == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \left (-\frac {8 c_1{}^3}{\sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^9 c_2{}^9 (-64+27 c_1 c_2)}-27 c_1{}^5 c_2{}^5}}+\frac {c_1{}^2}{c_2}-\frac {2 \sqrt [3]{\sqrt {3} \sqrt {c_1{}^9 c_2{}^9 (-64+27 c_1 c_2)}-9 c_1{}^5 c_2{}^5}}{3^{2/3} c_2{}^3}\right ) x^2+c_1 x+c_2\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(32*(x*Derivative(y(x), (x, 2)) - Derivative(y(x), x))**3*Derivative(y(x), (x, 2)) + (2*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve 32*(x*Derivative(y(x), (x, 2)) - Derivative(y(x), x))**3*D