23.4.213 problem 213

Internal problem ID [6515]
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 : 213
Date solved : Tuesday, September 30, 2025 at 03:02:34 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} a y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.297 (sec). Leaf size: 29

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

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

✗ Sympy

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