55.25.29 problem 29
Internal
problem
ID
[13738]
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
:
29
Date
solved
:
Sunday, October 12, 2025 at 05:13:59 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} \left (\left (a x +c \right ) y+\left (1-n \right ) x^{2}+\left (2 n -1\right ) x -n \right ) y^{\prime }&=2 a y^{2}+2 x y \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 484
ode:=((a*x+c)*y(x)+(-n+1)*x^2+(-1+2*n)*x-n)*diff(y(x),x) = 2*a*y(x)^2+2*x*y(x);
dsolve(ode,y(x), singsol=all);
\[
\frac {2 \left (a +c \right ) n \left (\operatorname {LegendreQ}\left (-\frac {2 n +1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right ) c_1 -\operatorname {LegendreP}\left (\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )\right ) \sqrt {\frac {y a^{2}}{n \left (a +c \right )}}-\left (a +c +\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}\right ) \left (\operatorname {LegendreQ}\left (-\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right ) c_1 -\operatorname {LegendreP}\left (-\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )\right )}{2 n \sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right ) \operatorname {LegendreQ}\left (-\frac {2 n +1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )-\left (a +c +\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}\right ) \operatorname {LegendreQ}\left (-\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )} = 0
\]
✗ Mathematica
ode=((a*x+c)*y[x]+(1-n)*x**2+(2*n-1)*x-n)*D[y[x],x]==2*a*y[x]^2+2*x*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Timed out
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
c = symbols("c")
n = symbols("n")
y = Function("y")
ode = Eq(-2*a*y(x)**2 - 2*x*y(x) + (-n + x**2*(1 - n) + x*(2*n - 1) + (a*x + c)*y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out