2.23.7 Problem 7
Internal
problem
ID
[13560]
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
:
Friday, December 19, 2025 at 06:57:23 AM
CAS
classification
:
[[_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&=a \,{\mathrm e}^{\lambda x} y+1 \\
\end{align*}
Unknown ode type.
2.23.7.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 83
ode:=y(x)*diff(y(x),x) = a*exp(lambda*x)*y(x)+1;
dsolve(ode,y(x), singsol=all);
\[
c_1 -\operatorname {erf}\left (\frac {\left (-\lambda y+a \,{\mathrm e}^{\lambda x}\right ) \sqrt {2}}{2 \sqrt {-\lambda }}\right ) \sqrt {2}\, \sqrt {\pi }\, a -2 \,{\mathrm e}^{\frac {{\mathrm e}^{2 \lambda x} a^{2}-2 a \lambda \,{\mathrm e}^{\lambda x} y-2 \left (-\frac {y^{2}}{2}+x \right ) \lambda ^{2}}{2 \lambda }} \sqrt {-\lambda } = 0
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=a \,{\mathrm e}^{\lambda x} y \left (x \right )+1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \,{\mathrm e}^{\lambda x} y \left (x \right )+1}{y \left (x \right )} \end {array} \]
2.23.7.2 ✓ 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]
\[
\text {Solve}\left [-\frac {a e^{\lambda x}}{\sqrt {\lambda }}=\frac {2 e^{\frac {\left (a e^{\lambda x}-\lambda y(x)\right )^2}{2 \lambda }}}{\sqrt {2 \pi } \text {erfi}\left (\frac {\lambda y(x)-a e^{\lambda x}}{\sqrt {2} \sqrt {\lambda }}\right )+2 c_1},y(x)\right ]
\]
2.23.7.3 ✗ 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