60.2.110 problem 686

Internal problem ID [10684]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 686
Date solved : Sunday, March 30, 2025 at 06:21:07 PM
CAS classification : [[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 47

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

\[ y = {\mathrm e}^{-x^{2}} \left (\cot \left (\operatorname {RootOf}\left (-2 x^{2}-\ln \left (2\right )+\ln \left (5\right )-\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (-1+\tan \left (\textit {\_Z} \right )\right )+6 c_1 -2 \textit {\_Z} \right )\right )-1\right ) \]

✓ Mathematica. Time used: 6.79 (sec). Leaf size: 97

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

\[ \text {Solve}\left [\int _1^{\frac {x \left (\frac {3}{e^{x^2} y(x)+1}-1\right )}{2^{2/3} \sqrt [3]{5} \sqrt [3]{-x^3}}}\frac {1}{K[1]^3-\frac {3 \sqrt [3]{-\frac {1}{2}} K[1]}{5^{2/3}}+1}dK[1]=\frac {1}{9} \sqrt [3]{2} 5^{2/3} \left (-x^3\right )^{2/3}+c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -x*y(x)**3*exp(2*x**2)/(y(x)*exp(x**2) + 1) + Derivative(y(x), x) cannot be solved by the lie group method