23.1.91 problem 85
Internal
problem
ID
[4698]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
85
Date
solved
:
Tuesday, September 30, 2025 at 08:15:25 AM
CAS
classification
:
[[_homogeneous, `class G`], _Abel]
\begin{align*} y^{\prime }&=\left (a +b x y\right ) y^{2} \end{align*}
✓ Maple. Time used: 0.119 (sec). Leaf size: 103
ode:=diff(y(x),x) = (a+b*x*y(x))*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 \sqrt {a^{2}-4 b}\, a \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{\textit {\_Z}}+a}{\sqrt {a^{2}-4 b}}\right )-\ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) a^{2}+2 c_1 \,a^{2}+2 \textit {\_Z} \,a^{2}+4 \ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) b -8 c_1 b -8 \textit {\_Z} b \right )}}{x}
\]
✓ Mathematica. Time used: 0.113 (sec). Leaf size: 94
ode=D[y[x],x]==(a+b*x*y[x])*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {a^2 \left (-\frac {2 \arctan \left (\frac {a+2 b x y(x)}{a \sqrt {\frac {4 b}{a^2}-1}}\right )}{\sqrt {\frac {4 b}{a^2}-1}}-\log \left (\frac {b x y(x) (a+b x y(x))+b}{b^2 x^2 y(x)^2}\right )\right )}{2 b}=\frac {a^2 \log (x)}{b}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 6.184 (sec). Leaf size: 335
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq((-a - b*x*y(x))*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - \frac {\left (- \frac {a}{\sqrt {a^{2} - 4 b}} - 1\right ) \log {\left (x y{\left (x \right )} + \frac {- \frac {a^{4} \left (- \frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )^{2}}{2} + 2 a^{4} + \frac {7 a^{2} b \left (- \frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )^{2}}{2} - \frac {3 a^{2} b \left (- \frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )}{2} - 11 a^{2} b - 6 b^{2} \left (- \frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )^{2} + 6 b^{2} \left (- \frac {a}{\sqrt {a^{2} - 4 b}} - 1\right ) + 12 b^{2}}{a b \left (2 a^{2} - 9 b\right )} \right )}}{2} - \frac {\left (\frac {a}{\sqrt {a^{2} - 4 b}} - 1\right ) \log {\left (x y{\left (x \right )} + \frac {- \frac {a^{4} \left (\frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )^{2}}{2} + 2 a^{4} + \frac {7 a^{2} b \left (\frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )^{2}}{2} - \frac {3 a^{2} b \left (\frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )}{2} - 11 a^{2} b - 6 b^{2} \left (\frac {a}{\sqrt {a^{2} - 4 b}} - 1\right )^{2} + 6 b^{2} \left (\frac {a}{\sqrt {a^{2} - 4 b}} - 1\right ) + 12 b^{2}}{a b \left (2 a^{2} - 9 b\right )} \right )}}{2} + \log {\left (x \right )} - \log {\left (x y{\left (x \right )} \right )} = 0
\]