54.2.373 problem 952

Internal problem ID [12247]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 952
Date solved : Sunday, October 12, 2025 at 02:27:37 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-y+\sqrt {y^{2}+x^{2}}\, x^{2}-x \sqrt {y^{2}+x^{2}}\, y+x^{4} \sqrt {y^{2}+x^{2}}-x^{3} \sqrt {y^{2}+x^{2}}\, y+x^{5} \sqrt {y^{2}+x^{2}}-x^{4} \sqrt {y^{2}+x^{2}}\, y}{x} \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 63

ode:=diff(y(x),x) = -(-y(x)+(x^2+y(x)^2)^(1/2)*x^2-x*(x^2+y(x)^2)^(1/2)*y(x)+x^4*(x^2+y(x)^2)^(1/2)-x^3*(x^2+y(x)^2)^(1/2)*y(x)+x^5*(x^2+y(x)^2)^(1/2)-x^4*(x^2+y(x)^2)^(1/2)*y(x))/x; 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (\frac {x \left (\sqrt {2 y^{2}+2 x^{2}}+y+x \right )}{-x +y}\right )+\frac {\left (4 x^{5}+5 x^{4}+10 x^{2}\right ) \sqrt {2}}{20}-c_1 +\ln \left (2\right )-\ln \left (x \right ) = 0 \]

✓ Mathematica. Time used: 60.179 (sec). Leaf size: 239

ode=D[y[x],x] == (y[x] - x^2*Sqrt[x^2 + y[x]^2] - x^4*Sqrt[x^2 + y[x]^2] - x^5*Sqrt[x^2 + y[x]^2] + x*y[x]*Sqrt[x^2 + y[x]^2] + x^3*y[x]*Sqrt[x^2 + y[x]^2] + x^4*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-8 \sqrt {x^2 \sinh ^6\left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{10 \sqrt {2}}\right ) \text {csch}^4\left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{5 \sqrt {2}}\right )}}{-1+2 \tanh ^2\left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{10 \sqrt {2}}\right )}\\ y(x)&\to \frac {x+8 \sqrt {x^2 \sinh ^6\left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{10 \sqrt {2}}\right ) \text {csch}^4\left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{5 \sqrt {2}}\right )}}{-1+2 \tanh ^2\left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{10 \sqrt {2}}\right )} \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**4*sqrt(x**2 + y(x)**2) - x**3*sqrt(x**2 + y(x)**2)*y(x) + x**2*sqrt(x**2 + y(x)**2) - x*sqrt(x**2 + y(x)**2)*y(x) - y(x))/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out