23.2.96 problem 98

Internal problem ID [5451]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 98
Date solved : Tuesday, September 30, 2025 at 12:43:32 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.011 (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: 0.028 (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: 0.425 (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)
 

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