60.1.113 problem 115

Internal problem ID [10127]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 115
Date solved : Sunday, March 30, 2025 at 03:18:49 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 50

ode:=x*diff(y(x),x)-x*(y(x)-x)*(x^2+y(x)^2)^(1/2)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (2\right )+\ln \left (\frac {x \left (\sqrt {2 y^{2}+2 x^{2}}+y+x \right )}{y-x}\right )+\frac {\sqrt {2}\, x^{2}}{2}-\ln \left (x \right )-c_1 = 0 \]

✓ Mathematica. Time used: 53.999 (sec). Leaf size: 154

ode=x*D[y[x],x] - x*(y[x]-x)*Sqrt[y[x]^2 + x^2] - y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {x-8 \sqrt {x^2 \sinh ^6\left (\frac {x^2+2 c_1}{\sqrt {2}}\right ) \text {csch}^4\left (\sqrt {2} \left (x^2+2 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {x^2+2 c_1}{\sqrt {2}}\right )} \\ y(x)\to \frac {x+8 \sqrt {x^2 \sinh ^6\left (\frac {x^2+2 c_1}{\sqrt {2}}\right ) \text {csch}^4\left (\sqrt {2} \left (x^2+2 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {x^2+2 c_1}{\sqrt {2}}\right )} \\ y(x)\to x \\ \end{align*}

✗ Sympy

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

Timed Out