23.2.117 problem 119

Internal problem ID [5472]
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 : 119
Date solved : Tuesday, September 30, 2025 at 12:44:12 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 111

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

\begin{align*} x -\frac {\left (-y+\sqrt {x^{2}+y^{2}}\right ) 2^{{1}/{3}} c_1}{2 x \left (\frac {3 y^{2}-3 y \sqrt {x^{2}+y^{2}}+x^{2}}{x^{2}}\right )^{{2}/{3}}} &= 0 \\ \frac {c_1 \left (y+\sqrt {x^{2}+y^{2}}\right )}{x {\left (\frac {3 y \sqrt {x^{2}+y^{2}}+x^{2}+3 y^{2}}{x^{2}}\right )}^{{2}/{3}}}+x &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.382 (sec). Leaf size: 6977

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

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✓ Sympy. Time used: 5.974 (sec). Leaf size: 75

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

\[ \left [ \log {\left (x \right )} = C_{1} - \log {\left (\left (\sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {2 y{\left (x \right )}}{x}\right )^{\frac {2}{3}} \right )} + \frac {\operatorname {asinh}{\left (\frac {y{\left (x \right )}}{x} \right )}}{3}, \ \log {\left (x \right )} = C_{1} - \log {\left (\left (- \sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {2 y{\left (x \right )}}{x}\right )^{\frac {2}{3}} \right )} - \frac {\operatorname {asinh}{\left (\frac {y{\left (x \right )}}{x} \right )}}{3}\right ] \]