54.2.393 problem 972

Internal problem ID [12267]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 972
Date solved : Wednesday, October 01, 2025 at 01:26:28 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {x \left (-x^{2}+2 x^{2} y-2 x^{4}+1\right )}{y-x^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=diff(y(x),x) = x*(-x^2+2*x^2*y(x)-2*x^4+1)/(-x^2+y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2}+\frac {\operatorname {LambertW}\left (-2 c_1 \,{\mathrm e}^{x^{4}-2 x^{2}-1}\right )}{2}+\frac {1}{2} \]

✓ Mathematica. Time used: 0.662 (sec). Leaf size: 43

ode=D[y[x],x] == (x*(1 - x^2 - 2*x^4 + 2*x^2*y[x]))/(-x^2 + y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2+\frac {1}{2} \left (1+W\left (-e^{x^4-2 x^2-1+c_1}\right )\right )\\ y(x)&\to x^2+\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.960 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(-2*x**4 + 2*x**2*y(x) - x**2 + 1)/(-x**2 + y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{2} + \frac {W\left (C_{1} e^{x^{4} - 2 x^{2} - 1}\right )}{2} + \frac {1}{2} \]