54.2.61 problem 637

Internal problem ID [11935]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 637
Date solved : Tuesday, September 30, 2025 at 11:45:30 PM
CAS classification : [[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 84

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

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

✓ Mathematica. Time used: 5.949 (sec). Leaf size: 130

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(-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*exp(-x**2)/(y(x)*exp(x**2) + 1) + Derivative(y(x), x) cannot be solved by the lie group method