83.10.18 problem 18

Internal problem ID [21987]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IV. First order differential equations of higher degree. Ex. XI at page 69
Problem number : 18
Date solved : Thursday, October 02, 2025 at 08:21:25 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

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

\begin{align*} y &= \frac {x^{2}}{2}+c_1 \\ y &= -\frac {x^{2}}{2}+c_1 \\ y &= c_1 \,{\mathrm e}^{-x} \\ \end{align*}

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to c_1 e^{-x}\\ y(x)&\to -\frac {x^2}{2}+c_1\\ y(x)&\to \frac {x^2}{2}+c_1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 24

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

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