60.1.431 problem 442

Internal problem ID [10445]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 442
Date solved : Sunday, March 30, 2025 at 04:43:42 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

ode:=x^2*diff(y(x),x)^2+(x^2*y(x)-2*x*y(x)+x^3)*diff(y(x),x)+(y(x)^2-x^2*y(x))*(1-x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (-x +c_1 \right ) x \\ y &= c_1 \,{\mathrm e}^{-x} x \\ \end{align*}

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 26

ode=(1 - x)*(-(x^2*y[x]) + y[x]^2) + (x^3 - 2*x*y[x] + x^2*y[x])*D[y[x],x] + x^2*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 e^{-x} x \\ y(x)\to x (-x+c_1) \\ \end{align*}

✓ Sympy. Time used: 0.376 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x)**2 + (1 - x)*(-x**2*y(x) + y(x)**2) + (x**3 + x**2*y(x) - 2*x*y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x \left (C_{1} - x\right ), \ y{\left (x \right )} = C_{1} x e^{- x}\right ] \]