83.10.15 problem 15

Internal problem ID [21984]
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 : 15
Date solved : Thursday, October 02, 2025 at 08:21:24 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Clairaut]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 31

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

\begin{align*} y &= -\frac {1}{4 x} \\ y &= c_1 x -\sqrt {c_1} \\ y &= c_1 x +\sqrt {c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.146 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to \frac {x-2 c_1}{4 c_1{}^2}\\ y(x)&\to \frac {x+2 c_1}{4 c_1{}^2}\\ y(x)&\to 0\\ y(x)&\to -\frac {1}{4 x} \end{align*}

✓ Sympy. Time used: 2.657 (sec). Leaf size: 10

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

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