23.2.84 problem 86

Internal problem ID [5439]
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 : 86
Date solved : Tuesday, September 30, 2025 at 12:43:22 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 77

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

\begin{align*} y &= \frac {x^{3}}{6} \\ y &= \frac {\sqrt {6}\, \sqrt {-c_1 x}\, x}{3}+c_1 \\ y &= -\frac {\sqrt {6}\, \sqrt {-c_1 x}\, x}{3}+c_1 \\ y &= -\frac {\sqrt {6}\, \sqrt {-c_1 x}\, x}{3}+c_1 \\ y &= \frac {\sqrt {6}\, \sqrt {-c_1 x}\, x}{3}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 1.378 (sec). Leaf size: 146

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

\begin{align*} \text {Solve}\left [\frac {1}{3} \log (y(x))-\frac {2 \sqrt {x^4-6 x y(x)} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x^3-6 y(x)}}\right )}{3 \sqrt {x} \sqrt {x^3-6 y(x)}}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {2 \sqrt {x^4-6 x y(x)} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x^3-6 y(x)}}\right )}{3 \sqrt {x} \sqrt {x^3-6 y(x)}}+\frac {1}{3} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to \frac {x^3}{6} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x**2/2 - sqrt(x*(x**3 - 6*y(x)))/2 + Derivative(y(x), x) cannot