60.2.286 problem 864

Internal problem ID [10860]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 864
Date solved : Wednesday, March 05, 2025 at 01:12:34 PM
CAS classification : [[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

ode:=diff(y(x),x) = y(x)*(exp(-1/4*x^2)^2*x*y(x)+exp(-1/4*x^2)*x+2*y(x)^2*exp(-3/4*x^2))*exp(1/4*x^2)/(2*y(x)*exp(-1/4*x^2)+2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{\sqrt {c_{1} -2 x}-1} \\ y &= \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{-\sqrt {c_{1} -2 x}-1} \\ \end{align*}

✓ Mathematica. Time used: 8.486 (sec). Leaf size: 103

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

\begin{align*} y(x)\to \frac {e^{\frac {x^2}{2}}}{-e^{\frac {x^2}{4}}+\sqrt {e^{\frac {x^2}{2}} (-2 x+1+c_1)}} \\ y(x)\to -\frac {e^{\frac {x^2}{2}}}{e^{\frac {x^2}{4}}+\sqrt {e^{\frac {x^2}{2}} (-2 x+1+c_1)}} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

Timed Out