55.26.7 problem 7
Internal
problem
ID
[13754]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.4.
Equations
Containing
Polynomial
Functions
of
y.
subsection
1.4.1-2
Abel
equations
of
the
first
kind.
Problem
number
:
7
Date
solved
:
Thursday, October 02, 2025 at 07:58:02 AM
CAS
classification
:
[[_homogeneous, `class G`], _Abel]
\begin{align*} y^{\prime }&=a x y^{3}+b y^{2} \end{align*}
✓ Maple. Time used: 0.119 (sec). Leaf size: 102
ode:=diff(y(x),x) = a*x*y(x)^3+b*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (-2 \sqrt {b^{2}-4 a}\, b \,\operatorname {arctanh}\left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}-4 a}}\right )+\ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) b^{2}-2 c_1 \,b^{2}-2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) a +8 c_1 a +8 \textit {\_Z} a \right )}}{x}
\]
✓ Mathematica. Time used: 0.11 (sec). Leaf size: 94
ode=D[y[x],x]==a*x*y[x]^3+b*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {b^2 \left (-\log \left (\frac {a x y(x) (a x y(x)+b)+a}{a^2 x^2 y(x)^2}\right )-\frac {2 \arctan \left (\frac {2 a x y(x)+b}{b \sqrt {\frac {4 a}{b^2}-1}}\right )}{\sqrt {\frac {4 a}{b^2}-1}}\right )}{2 a}=\frac {b^2 \log (x)}{a}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 5.705 (sec). Leaf size: 416
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*x*y(x)**3 - b*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - \frac {\left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) \log {\left (x y{\left (x \right )} + \frac {6 a^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2} - 6 a^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) - 12 a^{2} - \frac {7 a b^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} + \frac {3 a b^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )}{2} + 11 a b^{2} + \frac {b^{4} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} - 2 b^{4}}{a b \left (9 a - 2 b^{2}\right )} \right )}}{2} - \frac {\left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) \log {\left (x y{\left (x \right )} + \frac {6 a^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2} - 6 a^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) - 12 a^{2} - \frac {7 a b^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} + \frac {3 a b^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )}{2} + 11 a b^{2} + \frac {b^{4} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} - 2 b^{4}}{a b \left (9 a - 2 b^{2}\right )} \right )}}{2} + \log {\left (x \right )} - \log {\left (x y{\left (x \right )} \right )} = 0
\]