2.24.1 Problem 1
Internal
problem
ID
[13565]
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
:
1
Date
solved
:
Friday, December 19, 2025 at 07:01:48 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&=\left (x a +3 b \right ) y+x^{3} c -a b \,x^{2}-2 b^{2} x \\
\end{align*}
Unknown ode type.
2.24.1.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 201
ode:=y(x)*diff(y(x),x) = (a*x+3*b)*y(x)+x^3*c-a*b*x^2-2*b^2*x;
dsolve(ode,y(x), singsol=all);
\[
\frac {x \left (\frac {-a b \,x^{3}+c \,x^{4}+a \,x^{2} y-2 b^{2} x^{2}+4 b x y-2 y^{2}}{\left (-b x +y\right )^{2}}\right )^{{1}/{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {-a b x +2 c \,x^{2}+a y}{\left (-b x +y\right ) \sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}} y}{\sqrt {\frac {x^{2}}{-b x +y}}\, \left (-b x +y\right ) b}-\int _{}^{\frac {x^{2}}{-b x +y}}\frac {\left (\textit {\_a}^{2} c +\textit {\_a} a -2\right )^{{1}/{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} c +a}{\sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +c_1 = 0
\]
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
Looking for potential symmetries
found: 2 potential symmetries. Proceeding with integration step
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (a x +3 b \right ) y \left (x \right )+c \,x^{3}-a b \,x^{2}-2 b^{2} x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (a x +3 b \right ) y \left (x \right )+c \,x^{3}-a b \,x^{2}-2 b^{2} x}{y \left (x \right )} \end {array} \]
2.24.1.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==(a*x+3*b)*y[x]+c*x^3-a*b*x^2-2*b^2*x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.1.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*b*x**2 + 2*b**2*x - c*x**3 - (a*x + 3*b)*y(x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out