23.4.302 problem 305

Internal problem ID [6604]
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 : 305
Date solved : Tuesday, September 30, 2025 at 03:27:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}&=4 x y \left (-y+x y^{\prime }\right )^{3} \end{align*}

✗ Maple

ode:=(y(x)^2-x^2*diff(y(x),x)^2+x^2*y(x)*diff(diff(y(x),x),x))^2 = 4*x*y(x)*(-y(x)+x*diff(y(x),x))^3; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 76.152 (sec). Leaf size: 27

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

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

✗ Sympy

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

Timed Out