54.1.312 problem 318

Internal problem ID [11626]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 318
Date solved : Tuesday, September 30, 2025 at 09:49:08 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (3 x y^{3}-4 x y+y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right )&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=(3*x*y(x)^3-4*x*y(x)+y(x))*diff(y(x),x)+y(x)^2*(y(x)^2-2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ x +\frac {1}{y^{2}}-\frac {c_1}{y^{2} \sqrt {y^{2}-2}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.339 (sec). Leaf size: 161

ode=y[x]^2*(-2 + y[x]^2) + (y[x] - 4*x*y[x] + 3*x*y[x]^3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 0\\ \text {Solve}\left [x=\exp \left (\int _1^{y(x)}\frac {3 K[1]^2-4}{2 K[1]-K[1]^3}dK[1]\right ) \int _1^{y(x)}\frac {\exp \left (-\int _1^{K[2]}\frac {3 K[1]^2-4}{2 K[1]-K[1]^3}dK[1]\right )}{2 K[2]-K[2]^3}dK[2]+c_1 \exp \left (\int _1^{y(x)}\frac {3 K[1]^2-4}{2 K[1]-K[1]^3}dK[1]\right ),y(x)\right ]\\ y(x)&\to 0\\ y(x)&\to -\sqrt {2}\\ y(x)&\to \sqrt {2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)**2 - 2)*y(x)**2 + (3*x*y(x)**3 - 4*x*y(x) + y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out