54.2.296 problem 875

Internal problem ID [12170]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 875
Date solved : Sunday, October 12, 2025 at 02:25:51 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-x y-y+x^{5} \sqrt {y^{2}+x^{2}}-x^{4} \sqrt {y^{2}+x^{2}}\, y}{x \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 74

ode:=diff(y(x),x) = -(-x*y(x)-y(x)+x^5*(x^2+y(x)^2)^(1/2)-x^4*(x^2+y(x)^2)^(1/2)*y(x))/x/(1+x); 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (\frac {x \left (\sqrt {2 y^{2}+2 x^{2}}+y+x \right )}{-x +y}\right )+\sqrt {2}\, \ln \left (x +1\right )+\frac {\left (3 x^{4}-4 x^{3}+6 x^{2}-12 x \right ) \sqrt {2}}{12}-c_1 +\ln \left (2\right )-\ln \left (x \right ) = 0 \]

✓ Mathematica. Time used: 27.26 (sec). Leaf size: 232

ode=D[y[x],x] == (y[x] + x*y[x] - x^5*Sqrt[x^2 + y[x]^2] + x^4*y[x]*Sqrt[x^2 + y[x]^2])/(x*(1 + x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x-2 \sqrt {x^2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right ) \text {sech}^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )}}{-1+2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )}\\ y(x)&\to \frac {x+2 \sqrt {x^2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right ) \text {sech}^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )}}{-1+2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+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**5*sqrt(x**2 + y(x)**2) - x**4*sqrt(x**2 + y(x)**2)*y(x) - x*y(x) - y(x))/(x*(x + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out