83.23.8 problem 8

Internal problem ID [19218]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 8
Date solved : Monday, March 31, 2025 at 06:57:38 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.105 (sec). Leaf size: 77

ode:=3*y(x) = 2*x*diff(y(x),x)-2*diff(y(x),x)^2/x; 
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: 0.627 (sec). Leaf size: 146

ode=3*y[x]==2*D[y[x],x]*x-2*D[y[x],x]^2/x; 
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*Derivative(y(x), x) + 3*y(x) + 2*Derivative(y(x), x)**2/x,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 be solved by the factorable group method