23.2.196 problem 201

Internal problem ID [5551]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 201
Date solved : Tuesday, September 30, 2025 at 12:52:36 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.363 (sec). Leaf size: 87

ode:=y(x)*diff(y(x),x)^2+x^3*diff(y(x),x)-x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.541 (sec). Leaf size: 169

ode=y[x] (D[y[x],x])^2+x^3 D[y[x],x]-x^2 y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {\sqrt {x^4+4 y(x)^2}}{x^2}\right )}{2 x \sqrt {x^4+4 y(x)^2}}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {\sqrt {x^4+4 y(x)^2}}{x^2}\right )}{2 x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to -\frac {i x^2}{2}\\ y(x)&\to \frac {i x^2}{2} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x*(-x**2 + sqrt(x**4 + 4*y(x)**2))/(2*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method