54.1.396 problem 407

Internal problem ID [11710]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 407
Date solved : Tuesday, September 30, 2025 at 10:13:57 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

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

ode:=x*diff(y(x),x)^2-y(x) = 0; 
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=-y[x] + x*D[y[x],x]^2==0; 
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: 0.429 (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)
 

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