54.2.403 problem 982

Internal problem ID [12277]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 982
Date solved : Wednesday, October 01, 2025 at 01:26:53 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=\frac {y \,{\mathrm e}^{-\frac {x^{2}}{2}} \left (2 y^{2}+2 y \,{\mathrm e}^{\frac {x^{2}}{4}}+2 \,{\mathrm e}^{\frac {x^{2}}{2}}+x \,{\mathrm e}^{\frac {x^{2}}{2}}\right )}{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 76

ode:=diff(y(x),x) = 1/2*y(x)/exp(1/4*x^2)^2*(2*y(x)^2+2*y(x)*exp(1/4*x^2)+2*exp(1/4*x^2)^2+x*exp(1/4*x^2)^2); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {2 \ln \left (y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )}{3}-\frac {\ln \left (7\right )}{3}+\frac {\ln \left (1+y^{2} {\mathrm e}^{-\frac {x^{2}}{2}}+y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )}{3}+\frac {2 \sqrt {3}\, \arctan \left (\frac {2 \,{\mathrm e}^{-\frac {x^{2}}{4}} y \sqrt {3}}{3}+\frac {\sqrt {3}}{3}\right )}{9}+\frac {2 x}{3}-c_1 = 0 \]

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 113

ode=D[y[x],x] == (y[x]*(2*E^(x^2/2) + E^(x^2/2)*x + 2*E^(x^2/4)*y[x] + 2*y[x]^2))/(2*E^(x^2/2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-\frac {x^2}{2}} y(x)+e^{-\frac {x^2}{4}}}{\sqrt [3]{7} \sqrt [3]{-e^{-\frac {3 x^2}{4}}}}}\frac {1}{K[1]^3-\frac {6 \sqrt [3]{-1} K[1]}{7^{2/3}}+1}dK[1]=\frac {1}{9} 7^{2/3} e^{\frac {x^2}{2}} \left (-e^{-\frac {3 x^2}{4}}\right )^{2/3} x+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x*exp(x**2/2) - 2*y(x)**2 - 2*y(x)*exp(x**2/4) - 2*exp(x**2/2))*y(x)*exp(-x**2/2)/2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -((x*exp(x**2/2) + 2*y(x)**2 + 2*exp(x**2/2))*exp(x**2)**(1/4) + 2*y(x)*sqrt(exp(x**2)))*y(x)/(2*exp(x**2)**(3/4)) + Derivative(y(x), x) cannot be solved by the factorable group method