23.4.226 problem 226

Internal problem ID [6528]
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 : 226
Date solved : Friday, October 03, 2025 at 02:09:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 38

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

\begin{align*} y &= x \\ y &= -\operatorname {RootOf}\left (\textit {\_Z}^{a +1} x^{-a}-c_2 a x +c_1 a -c_2 x +c_1 \right )+x \\ \end{align*}

✓ Mathematica. Time used: 60.216 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to x \left (1+\left (\frac {(-1)^{-a} (a+1) \left ((-1)^a c_2 x+c_1\right )}{x}\right ){}^{\frac {1}{a+1}}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (a*y(x) + x*sqrt(a*(x - y(x))*Derivative(y