2.24.44 Problem 70
Internal
problem
ID
[13608]
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
:
70
Date
solved
:
Friday, December 19, 2025 at 08:29:16 AM
CAS
classification
:
[[_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&={\mathrm e}^{\lambda x} \left (2 x a \lambda +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \\
\end{align*}
Unknown ode type.
2.24.44.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 118
ode:=y(x)*diff(y(x),x) = exp(lambda*x)*(2*a*lambda*x+a+b)*y(x)-exp(2*lambda*x)*(a^2*lambda*x^2+a*b*x+c);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\tan \left (\frac {\operatorname {RootOf}\left (2 a x \lambda \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}\, \tan \left (\frac {\textit {\_a} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) \textit {\_Z} a +b \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 a \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}}{2}\right ) \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}\, a +2 a \lambda x +b \right ) {\mathrm e}^{\lambda x}}{2 \lambda }
\]
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
Looking for potential symmetries
Looking for potential symmetries
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )={\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y \left (x \right )-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {{\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y \left (x \right )-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )}{y \left (x \right )} \end {array} \]
2.24.44.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==Exp[\[Lambda]*x]*(2*a*\[Lambda]*x+a+b)*y[x]-Exp[2*\[Lambda]*x]*(a^2*\[Lambda]*x^2+a*b*x+c);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.44.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq((-2*a*lambda_*x - a - b)*y(x)*exp(lambda_*x) + (a**2*lambda_*x**2 + a*b*x + c)*exp(2*lambda_*x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0