27.1.2 problem problem 41
Internal
problem
ID
[7155]
Book
:
Differential
Gleichungen,
Kamke,
3rd
ed,
Abel
ODEs
Section
:
Abel
ODE
with
constant
invariant
Problem
number
:
problem
41
Date
solved
:
Tuesday, September 30, 2025 at 04:23:30 PM
CAS
classification
:
[[_homogeneous, `class G`], _Abel]
\begin{align*} a x y^{3}+b y^{2}+y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.117 (sec). Leaf size: 103
ode:=a*x*y(x)^3+b*y(x)^2+diff(y(x),x) = 0;
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: 0.112 (sec). Leaf size: 103
ode=a*x*y[x]^3+b*y[x]^2+D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-\frac {b^2 \left (\frac {2 \arctan \left (\frac {-2 a x y(x)-b}{b \sqrt {-\frac {4 a}{b^2}-1}}\right )}{\sqrt {-\frac {4 a}{b^2}-1}}-\log \left (\frac {a (-x) y(x) (-a x y(x)-b)-a}{a^2 x^2 y(x)^2}\right )\right )}{2 a}=-\frac {b^2 \log (x)}{a}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 6.045 (sec). Leaf size: 325
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)
\[
C_{1} + \frac {\left (- \frac {b}{\sqrt {4 a + b^{2}}} + 1\right ) \log {\left (x y{\left (x \right )} + \frac {- 6 a^{2} \left (- \frac {b}{\sqrt {4 a + b^{2}}} + 1\right )^{2} - 6 a^{2} \left (- \frac {b}{\sqrt {4 a + b^{2}}} + 1\right ) + 12 a^{2} - \frac {7 a b^{2} \left (- \frac {b}{\sqrt {4 a + b^{2}}} + 1\right )^{2}}{2} - \frac {3 a b^{2} \left (- \frac {b}{\sqrt {4 a + b^{2}}} + 1\right )}{2} + 11 a b^{2} - \frac {b^{4} \left (- \frac {b}{\sqrt {4 a + b^{2}}} + 1\right )^{2}}{2} + 2 b^{4}}{a b \left (9 a + 2 b^{2}\right )} \right )}}{2} + \frac {\left (\frac {b}{\sqrt {4 a + b^{2}}} + 1\right ) \log {\left (x y{\left (x \right )} + \frac {- 6 a^{2} \left (\frac {b}{\sqrt {4 a + b^{2}}} + 1\right )^{2} - 6 a^{2} \left (\frac {b}{\sqrt {4 a + b^{2}}} + 1\right ) + 12 a^{2} - \frac {7 a b^{2} \left (\frac {b}{\sqrt {4 a + b^{2}}} + 1\right )^{2}}{2} - \frac {3 a b^{2} \left (\frac {b}{\sqrt {4 a + b^{2}}} + 1\right )}{2} + 11 a b^{2} - \frac {b^{4} \left (\frac {b}{\sqrt {4 a + b^{2}}} + 1\right )^{2}}{2} + 2 b^{4}}{a b \left (9 a + 2 b^{2}\right )} \right )}}{2} + \log {\left (x \right )} - \log {\left (x y{\left (x \right )} \right )} = 0
\]