83.10.4 problem 4

Internal problem ID [21973]
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 : 4
Date solved : Thursday, October 02, 2025 at 08:20:54 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 23

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

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

✓ Sympy. Time used: 0.931 (sec). Leaf size: 10

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

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