60.1.332 problem 339

Internal problem ID [10346]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 339
Date solved : Sunday, March 30, 2025 at 04:18:20 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} \left (x \sqrt {x^{2}+y^{2}+1}-y \left (y^{2}+x^{2}\right )\right ) y^{\prime }-y \sqrt {x^{2}+y^{2}+1}-x \left (y^{2}+x^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 25

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

\[ \arctan \left (\frac {x}{y}\right )+\sqrt {x^{2}+y^{2}+1}-c_1 = 0 \]

✓ Mathematica. Time used: 0.52 (sec). Leaf size: 661

ode=-(x*(x^2 + y[x]^2)) - y[x]*Sqrt[1 + x^2 + y[x]^2] + (-(y[x]*(x^2 + y[x]^2)) + x*Sqrt[1 + x^2 + y[x]^2])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^3+\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) x^2 K[3]-\int _1^x\left (2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] \left (-\frac {1}{2 \left (K[2]^2+K[3]^2+1\right )}-\frac {1}{K[2]^2+K[3]^2}\right ) K[2]^3+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] K[2]+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^3 \left (-\frac {1}{2 \left (K[2]^2+K[3]^2+1\right )}-\frac {1}{K[2]^2+K[3]^2}\right ) K[2]+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 \sqrt {K[2]^2+K[3]^2+1} \left (-\frac {1}{2 \left (K[2]^2+K[3]^2+1\right )}-\frac {1}{K[2]^2+K[3]^2}\right )+\exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {K[2]^2+K[3]^2+1}+\frac {\exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2}{\sqrt {K[2]^2+K[3]^2+1}}\right )dK[2]-\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) x \sqrt {x^2+K[3]^2+1}\right )dK[3]+\int _1^x\left (\exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[2]^3+\exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) y(x)^2 K[2]+\exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) y(x) \sqrt {K[2]^2+y(x)^2+1}\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x**2 + y(x)**2) + (x*sqrt(x**2 + y(x)**2 + 1) - (x**2 + y(x)**2)*y(x))*Derivative(y(x), x) - sqrt(x**2 + y(x)**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out