83.10.20 problem 20

Internal problem ID [21989]
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 : 20
Date solved : Thursday, October 02, 2025 at 08:21:28 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 39

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

\begin{align*} y &= 0 \\ y &= \frac {\left (x +\sqrt {c_1 x}\right )^{2}}{x} \\ y &= \frac {\left (-x +\sqrt {c_1 x}\right )^{2}}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 46

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

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

✓ Sympy. Time used: 0.421 (sec). Leaf size: 15

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

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