54.2.398 problem 977

Internal problem ID [12272]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 977
Date solved : Wednesday, October 01, 2025 at 01:26:38 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], _Abel]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 85

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

\[ y = \frac {\left (\sqrt {11}\, \tan \left (\operatorname {RootOf}\left (-4 \sqrt {11}\, x^{2}-8 \sqrt {11}\, \ln \left (11\right )+8 \sqrt {11}\, \ln \left (-\sqrt {11}+11 \tan \left (\textit {\_Z} \right )\right )-4 \sqrt {11}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} {\mathrm e}^{2 x^{2}}\right )+8 \sqrt {11}\, \ln \left (5\right )+9 \sqrt {11}\, c_1 -8 \textit {\_Z} \right )\right )-1\right ) {\mathrm e}^{-x^{2}}}{2} \]

✓ Mathematica. Time used: 0.198 (sec). Leaf size: 109

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

\[ \text {Solve}\left [\int _1^{\frac {e^{x^2} x+3 e^{2 x^2} y(x) x}{5^{2/3} \sqrt [3]{-e^{3 x^2} x^3}}}\frac {1}{K[1]^3-\frac {24}{5} \sqrt [3]{-\frac {1}{5}} K[1]+1}dK[1]=-\frac {5 \sqrt [3]{5} e^{x^2} x^3}{18 \sqrt [3]{-e^{3 x^2} x^3}}+c_1,y(x)\right ] \]

✗ Sympy

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

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