54.2.248 problem 825

Internal problem ID [12122]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 825
Date solved : Sunday, October 12, 2025 at 02:21:06 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=\frac {\left (\left (x^{2}+1\right )^{{3}/{2}} x^{2}+\left (x^{2}+1\right )^{{3}/{2}}+y^{2} \left (x^{2}+1\right )^{{3}/{2}}+x^{2} y^{3}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}} \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 48

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

\[ y = \frac {\sqrt {x^{2}+1}\, \left (19 \operatorname {RootOf}\left (-1296 \int _{}^{\textit {\_Z}}\frac {1}{361 \textit {\_a}^{3}-432 \textit {\_a} +432}d \textit {\_a} +2 \ln \left (x^{2}+1\right )+3 c_1 \right )-6\right )}{18} \]

✓ Mathematica. Time used: 1.308 (sec). Leaf size: 129

ode=D[y[x],x] == (x*((1 + x^2)^(3/2) + x^2*(1 + x^2)^(3/2) + (1 + x^2)^(3/2)*y[x]^2 + y[x]^3 + x^2*y[x]^3))/(1 + x^2)^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {3 y(x) x}{\left (x^2+1\right )^2}+\frac {x}{\left (x^2+1\right )^{3/2}}}{\sqrt [3]{38} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^{9/2}}}}}\frac {1}{K[1]^3-\frac {6 \sqrt [3]{2} K[1]}{19^{2/3}}+1}dK[1]=\frac {19^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^{9/2}}\right )^{2/3} \left (x^2+1\right )^3 \log \left (x^2+1\right )}{9 \sqrt [3]{2} x^2}+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x**2*(x**2 + 1)**(3/2) + x**2*y(x)**3 + (x**2 + 1)**(3/2)*y(x)**2 + (x**2 + 1)**(3/2) + y(x)**3)/(x**2 + 1)**3 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out