2.25.5 Problem 7
Internal
problem
ID
[13620]
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
1.3.4-2.
Problem
number
:
7
Date
solved
:
Friday, December 19, 2025 at 08:53:57 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y^{\prime } y x&=-n y^{2}+a \left (1+2 n \right ) x y+b y-a^{2} n \,x^{2}-a b x +c \\
\end{align*}
Unknown ode type.
2.25.5.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 224
ode:=x*y(x)*diff(y(x),x) = -n*y(x)^2+a*(1+2*n)*x*y(x)+b*y(x)-a^2*n*x^2-a*b*x+c;
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (\frac {1}{x a -y}\right )^{\frac {1}{n}} \left (\frac {-n y^{2}+\left (2 a n x +b \right ) y-a^{2} n \,x^{2}-a b x +c}{\left (x a -y\right )^{2}}\right )^{-\frac {1}{2 n}} y \,{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-a b x +b y+2 c}{\sqrt {b^{2}+4 c n}\, \left (-x a +y\right )}\right )}{\sqrt {b^{2}+4 c n}\, n}}-\left (x a -y\right ) \left (\int _{}^{\frac {1}{x a -y}}\textit {\_a}^{\frac {1}{n}} \left (\textit {\_a}^{2} c -\textit {\_a} b -n \right )^{-\frac {1}{2 n}} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-2 \textit {\_a} c +b}{\sqrt {b^{2}+4 c n}}\right )}{n \sqrt {b^{2}+4 c n}}}d \textit {\_a} a -c_1 \right ) x}{\left (x a -y\right ) x} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=-13620 y \left (x \right )^{2}+27241 a x y \left (x \right )+b y \left (x \right )-13620 a^{2} x^{2}-a b x +c \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-13620 y \left (x \right )^{2}+27241 a x y \left (x \right )+b y \left (x \right )-13620 a^{2} x^{2}-a b x +c}{x y \left (x \right )} \end {array} \]
2.25.5.2 ✗ Mathematica
ode=x*y[x]*D[y[x],x]==-n*y[x]^2+a*(2*n+1)*x*y[x]+b*y[x]-a^2*n*x^2-a*b*x+c;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.25.5.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
n = symbols("n")
y = Function("y")
ode = Eq(a**2*n*x**2 + a*b*x - a*x*(2*n + 1)*y(x) - b*y(x) - c + n*y(x)**2 + x*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0