60.1.37 problem 37

Internal problem ID [10051]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 37
Date solved : Sunday, March 30, 2025 at 02:55:42 PM
CAS classification : [_Abel]

\begin{align*} y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 76

ode:=diff(y(x),x)-y(x)^3-a*exp(x)*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\operatorname {erf}\left (\frac {\left ({\mathrm e}^{x} a y+1\right ) \sqrt {2}}{2 y}\right ) \sqrt {2}\, \sqrt {\pi }\, a +2 c_1 a +2 \,{\mathrm e}^{-\frac {y^{2} {\mathrm e}^{2 x} a^{2}+2 x y^{2}+2 \,{\mathrm e}^{x} a y+1}{2 y^{2}}}}{2 a} = 0 \]

✓ Mathematica. Time used: 0.685 (sec). Leaf size: 78

ode=D[y[x],x] - y[x]^3 - a*Exp[x]*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-i a e^x=\frac {2 e^{\frac {1}{2} \left (-i a e^x-\frac {i}{y(x)}\right )^2}}{\sqrt {2 \pi } \text {erfi}\left (\frac {-i a e^x-\frac {i}{y(x)}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]

✗ Sympy

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

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