54.2.141 problem 718

Internal problem ID [12015]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 718
Date solved : Tuesday, September 30, 2025 at 11:55:43 PM
CAS classification : [_Abel]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 49

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

\[ y = -\frac {11 \,{\mathrm e}^{-x^{2}} \operatorname {RootOf}\left (-5 x^{2}+20250 \int _{}^{\textit {\_Z}}\frac {1}{121 \textit {\_a}^{3}+3375 \textit {\_a} -3375}d \textit {\_a} +6 c_1 \right )}{45}-\frac {{\mathrm e}^{-x^{2}}}{3} \]

✓ Mathematica. Time used: 0.191 (sec). Leaf size: 103

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

\[ \text {Solve}\left [\int _1^{\frac {e^{x^2} x+3 e^{2 x^2} y(x) x}{\sqrt [3]{11} \sqrt [3]{e^{3 x^2} x^3}}}\frac {1}{K[1]^3+\frac {15 K[1]}{11^{2/3}}+1}dK[1]=\frac {11^{2/3} e^{x^2} x^3}{18 \sqrt [3]{e^{3 x^2} x^3}}+c_1,y(x)\right ] \]

✗ Sympy

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

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