23.4.222 problem 222

Internal problem ID [6524]
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 : 222
Date solved : Friday, October 03, 2025 at 02:09:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 91

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

\begin{align*} y &= 0 \\ y &= {\left (-\frac {\left (4 c a -b^{2}+2 b +4 c -1\right ) x^{\sqrt {\left (-4 a -4\right ) c +b^{2}-2 b +1}+b -1}}{\left (a +1\right )^{2} \left (c_1 \,x^{\sqrt {\left (-4 a -4\right ) c +b^{2}-2 b +1}}-c_2 \right )^{2}}\right )}^{-\frac {1}{2 a +2}} \\ \end{align*}

✓ Mathematica. Time used: 15.815 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to c_2 x^{-\frac {\sqrt {-4 (a+1) c+b^2-2 b+1}+b-1}{2 (a+1)}} \left (x^{\sqrt {-4 (a+1) c+b^2-2 b+1}}+c_1\right ){}^{\frac {1}{a+1}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x) + sqrt((-4*a*c*y(x) - 4*a*x**2*De