23.4.113 problem 113

Internal problem ID [6415]
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 : 113
Date solved : Friday, October 03, 2025 at 02:05:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 37

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

\begin{align*} y &= \left (-\arctan \left (\frac {1}{\sqrt {c_1 \,x^{2}-1}}\right )+c_2 \right ) x \\ y &= \left (\arctan \left (\frac {1}{\sqrt {c_1 \,x^{2}-1}}\right )+c_2 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 54.062 (sec). Leaf size: 161

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

\begin{align*} y(x)&\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right )\\ y(x)&\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right )\\ y(x)&\to -i x \log \left (-\frac {i x}{\sqrt {2} \sqrt {-i c_1 x^2}}\right )\\ y(x)&\to -i x \log \left (\frac {i x}{\sqrt {2} \sqrt {-i c_1 x^2}}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(-(x**4*Derivative(y(x), (x, 2)) - y(x)**3)/x**3 - y(x)**3/x**3