79.2.8 problem (i)

Internal problem ID [21090]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises XIII at page 24
Problem number : (i)
Date solved : Thursday, October 02, 2025 at 07:07:19 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 26

ode:=y(x) = x*diff(y(x),x)-(x^2+y(x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-c_1 \,x^{2}+\sqrt {x^{2}+y^{2}}+y}{x^{2}} = 0 \]

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to x \sinh (\log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.733 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = - x \sinh {\left (C_{1} - \log {\left (x \right )} \right )} \]