54.2.250 problem 827

Internal problem ID [12124]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 827
Date solved : Sunday, October 12, 2025 at 02:21:40 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-y+x^{3} \sqrt {y^{2}+x^{2}}-x^{2} \sqrt {y^{2}+x^{2}}\, y}{x} \end{align*}

✓ Maple. Time used: 0.162 (sec). Leaf size: 50

ode:=diff(y(x),x) = -(-y(x)+x^3*(x^2+y(x)^2)^(1/2)-x^2*(x^2+y(x)^2)^(1/2)*y(x))/x; 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (2\right )+\ln \left (\frac {x \left (\sqrt {2 y^{2}+2 x^{2}}+y+x \right )}{-x +y}\right )+\frac {\sqrt {2}\, x^{3}}{3}-\ln \left (x \right )-c_1 = 0 \]

✓ Mathematica. Time used: 36.779 (sec). Leaf size: 172

ode=D[y[x],x] == (y[x] - x^3*Sqrt[x^2 + y[x]^2] + x^2*y[x]*Sqrt[x^2 + y[x]^2])/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x-2 \sqrt {x^2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right ) \text {sech}^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )}\\ y(x)&\to \frac {x+2 \sqrt {x^2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right ) \text {sech}^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )}\\ y(x)&\to x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + (x**3*sqrt(x**2 + y(x)**2) - x**2*sqrt(x**2 + y(x)**2)*y(x) - y(x))/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out