55.26.5 problem 5
Internal
problem
ID
[13752]
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
:
5
Date
solved
:
Thursday, October 02, 2025 at 07:57:50 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Abel]
\begin{align*} y^{\prime }&=-y^{3}+\frac {y^{2}}{\sqrt {a x +b}} \end{align*}
✓ Maple. Time used: 0.287 (sec). Leaf size: 148
ode:=diff(y(x),x) = -y(x)^3+1/(a*x+b)^(1/2)*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
\frac {\ln \left (\left (\left (2 a x +2 b \right ) y^{2}-a \right ) \sqrt {a x +b}-2 y \left (a x +b \right )\right ) \sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}+\left (2 a x +2 b \right ) \operatorname {arctanh}\left (\frac {\left (a x +b \right ) \left (2 \sqrt {a x +b}\, y-1\right )}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right )-\left (c_1 +2 \ln \left (y\right )+\frac {\ln \left (a x +b \right )}{2}\right ) \sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}} = 0
\]
✓ Mathematica. Time used: 0.14 (sec). Leaf size: 108
ode=D[y[x],x]==-y[x]^3+(a*x+b)^(-1/2)*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-\frac {-\frac {2 \arctan \left (\frac {\sqrt {a x+b} \left (\frac {1}{\sqrt {a x+b}}-2 y(x)\right )}{\sqrt {-2 a-1}}\right )}{\sqrt {-2 a-1}}-\log \left (\frac {-y(x) \left (\frac {1}{\sqrt {a x+b}}-y(x)\right )-\frac {a}{2 (a x+b)}}{y(x)^2}\right )}{a}=-\frac {\log (a x+b)}{a}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 19.924 (sec). Leaf size: 282
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(y(x)**3 + Derivative(y(x), x) - y(x)**2/sqrt(a*x + b),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - \frac {\left (1 - \frac {1}{\sqrt {2 a + 1}}\right ) \log {\left (\sqrt {a x + b} y{\left (x \right )} + \frac {3 a^{2} \left (1 - \frac {1}{\sqrt {2 a + 1}}\right )^{2} + 3 a^{2} \left (1 - \frac {1}{\sqrt {2 a + 1}}\right ) - 6 a^{2} + \frac {7 a \left (1 - \frac {1}{\sqrt {2 a + 1}}\right )^{2}}{2} + \frac {3 a \left (1 - \frac {1}{\sqrt {2 a + 1}}\right )}{2} - 11 a + \left (1 - \frac {1}{\sqrt {2 a + 1}}\right )^{2} - 4}{9 a + 4} \right )}}{2} - \frac {\left (1 + \frac {1}{\sqrt {2 a + 1}}\right ) \log {\left (\sqrt {a x + b} y{\left (x \right )} + \frac {3 a^{2} \left (1 + \frac {1}{\sqrt {2 a + 1}}\right )^{2} + 3 a^{2} \left (1 + \frac {1}{\sqrt {2 a + 1}}\right ) - 6 a^{2} + \frac {7 a \left (1 + \frac {1}{\sqrt {2 a + 1}}\right )^{2}}{2} + \frac {3 a \left (1 + \frac {1}{\sqrt {2 a + 1}}\right )}{2} - 11 a + \left (1 + \frac {1}{\sqrt {2 a + 1}}\right )^{2} - 4}{9 a + 4} \right )}}{2} + \log {\left (\sqrt {a x + b} y{\left (x \right )} \right )} - \frac {\log {\left (2 a x + 2 b \right )}}{2} = 0
\]