23.4.221 problem 221

Internal problem ID [6523]
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 : 221
Date solved : Friday, October 03, 2025 at 02:09:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y y^{\prime \prime }&=a y^{2}+a x y y^{\prime }+2 x^{2} {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 24

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

\begin{align*} y &= 0 \\ y &= \frac {a -1}{x^{a} c_1 -c_2 x} \\ \end{align*}

✓ Mathematica. Time used: 1.284 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to \frac {a-1}{(a-1) c_2 x^a-c_1 x}\\ y(x)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x*y(x)*Derivative(y(x), x) - a*y(x)**2 + x**2*y(x)*Derivative(y(x), (x, 2)) - 2*x**2*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) + sqrt((a**2*y(x) - 8*a*y(x) + 8*