61.24.67 problem 67

Internal problem ID [12401]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2.
Problem number : 67
Date solved : Monday, March 31, 2025 at 05:30:02 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 153

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

\[ {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {-2 \,{\mathrm e}^{x} c +a \left (b -y\right )}{\sqrt {a^{2}+4 c}\, \left (b -y\right )}\right )}{\sqrt {a^{2}+4 c}}} \sqrt {\frac {c \,{\mathrm e}^{2 x}-\left (b -y\right ) \left (a \,{\mathrm e}^{x}+b -y\right )}{\left (b -y\right )^{2}}}\, y-\int _{}^{\frac {{\mathrm e}^{x}}{-b +y}}\frac {\sqrt {\textit {\_a}^{2} c +\textit {\_a} a -1}\, {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} c +a}{\sqrt {a^{2}+4 c}}\right )}{\sqrt {a^{2}+4 c}}}}{\textit {\_a}}d \textit {\_a} b +c_1 = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]==(a*Exp[x]+b)*y[x]+c*Exp[2*x]-a*b*Exp[x]-b^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

Timed Out