23.1.85 problem 79

Internal problem ID [4692]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 79
Date solved : Sunday, October 12, 2025 at 01:17:54 AM
CAS classification : [_Abel]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 76

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

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

✓ Mathematica. Time used: 0.399 (sec). Leaf size: 78

ode=D[y[x],x]==(a*Exp[x]+y[x])*y[x]^2; 
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*exp(x) - y(x))*y(x)**2 + 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