60.2.368 problem 946

Internal problem ID [10942]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 946
Date solved : Sunday, March 30, 2025 at 07:27:07 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 99

ode:=diff(y(x),x) = (-8*exp(-x^2)*y(x)+4*x^2*exp(-x^2)^2-8*exp(-x^2)+8*x^2*exp(-x^2)*y(x)-4*x^4*exp(-x^2)^2+8*x^2*exp(-x^2)-8*y(x)^3+12*x^2*exp(-x^2)*y(x)^2-6*y(x)*x^4*exp(-x^2)^2+x^6*exp(-x^2)^3)*x/(-8*y(x)+4*x^2*exp(-x^2)-8); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{2} {\mathrm e}^{-x^{2}}}{2} \\ y &= \frac {2+x^{2} \left (\sqrt {-x^{2}+c_1}-1\right ) {\mathrm e}^{-x^{2}}}{2 \sqrt {-x^{2}+c_1}-2} \\ y &= \frac {-2+x^{2} \left (\sqrt {-x^{2}+c_1}+1\right ) {\mathrm e}^{-x^{2}}}{2 \sqrt {-x^{2}+c_1}+2} \\ \end{align*}

✓ Mathematica. Time used: 1.117 (sec). Leaf size: 93

ode=D[y[x],x] == (x*(-8/E^x^2 + (4*x^2)/E^(2*x^2) + (8*x^2)/E^x^2 - (4*x^4)/E^(2*x^2) + x^6/E^(3*x^2) - (8*y[x])/E^x^2 + (8*x^2*y[x])/E^x^2 - (6*x^4*y[x])/E^(2*x^2) + (12*x^2*y[x]^2)/E^x^2 - 8*y[x]^3))/(-8 + (4*x^2)/E^x^2 - 8*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{2} e^{-x^2} x^2+\frac {8}{-8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2-\frac {8}{8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2 \\ \end{align*}

✓ Sympy. Time used: 7.162 (sec). Leaf size: 116

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x**6*exp(-3*x**2) - 6*x**4*y(x)*exp(-2*x**2) - 4*x**4*exp(-2*x**2) + 12*x**2*y(x)**2*exp(-x**2) + 8*x**2*y(x)*exp(-x**2) + 8*x**2*exp(-x**2) + 4*x**2*exp(-2*x**2) - 8*y(x)**3 - 8*y(x)*exp(-x**2) - 8*exp(-x**2))/(4*x**2*exp(-x**2) - 8*y(x) - 8) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\left (\sqrt {- \left (C_{1} + x^{2} - 1\right ) e^{4 x^{2}}} + \frac {\left (C_{1} x^{2} + x^{4} - 2 e^{x^{2}}\right ) e^{x^{2}}}{2}\right ) e^{- 2 x^{2}}}{C_{1} + x^{2}}, \ y{\left (x \right )} = \frac {\left (C_{1} x^{2} e^{x^{2}} + \frac {x^{4} e^{x^{2}}}{2} - \sqrt {\left (- 2 C_{1} - x^{2} + 1\right ) e^{4 x^{2}}} - e^{2 x^{2}}\right ) e^{- 2 x^{2}}}{2 C_{1} + x^{2}}\right ] \]