83.22.13 problem 13

Internal problem ID [19196]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 13
Date solved : Monday, March 31, 2025 at 06:53:39 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.350 (sec). Leaf size: 87

ode:=x^2*(y(x)-x*diff(y(x),x)) = y(x)*diff(y(x),x)^2; 
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: 0.687 (sec). Leaf size: 169

ode=x^2*(y[x]-D[y[x],x]*x)==y[x]*D[y[x],x]^2; 
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**2*(-x*Derivative(y(x), x) + 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