25.2.8 problem 8.1

Internal problem ID [6864]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 3
Problem number : 8.1
Date solved : Tuesday, September 30, 2025 at 03:56:08 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact]

\begin{align*} \frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 661

ode= x/Sqrt[1+x^2+y[x]^2] + y[x]/Sqrt[1+x^2+y[x]^2]*D[y[x],x]+y[x]/(x^2+y[x]^2) - x/(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/sqrt(x**2 + y(x)**2 + 1) - x*Derivative(y(x), x)/(x**2 + y(x)**2) + y(x)*Derivative(y(x), x)/sqrt(x**2 + y(x)**2 + 1) + y(x)/(x**2 + y(x)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out