55.23.7 problem 7

Internal problem ID [13624]
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 1.3.2.
Problem number : 7
Date solved : Sunday, October 12, 2025 at 04:15:27 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=a \,{\mathrm e}^{\lambda x} y+1 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 83

ode:=y(x)*diff(y(x),x) = a*exp(lambda*x)*y(x)+1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.706 (sec). Leaf size: 83

ode=y[x]*D[y[x],x]==a*Exp[\[Lambda]*x]*y[x]+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-a*y(x)*exp(lambda_*x) + y(x)*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out