54.1.289 problem 295

Internal problem ID [11603]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 295
Date solved : Tuesday, September 30, 2025 at 09:37:42 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.105 (sec). Leaf size: 29

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} c_1 +\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+1\right )} x \]

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 31

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

\[ \text {Solve}\left [\frac {x}{y(x)}+\frac {y(x)}{x}+\log \left (\frac {y(x)}{x}\right )=-2 \log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.854 (sec). Leaf size: 27

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

\[ \log {\left (x \right )} = C_{1} - \frac {x}{2 y{\left (x \right )}} - \log {\left (\sqrt {\frac {y{\left (x \right )}}{x}} \right )} - \frac {y{\left (x \right )}}{2 x} \]