54.1.213 problem 217

Internal problem ID [11527]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 217
Date solved : Tuesday, September 30, 2025 at 08:57:08 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 2.397 (sec). Leaf size: 40

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

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

✓ Sympy. Time used: 0.553 (sec). Leaf size: 24

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

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