2.25.15 Problem 29

Internal problem ID [13630]
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 : Friday, December 19, 2025 at 09:12:06 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (\left (x a +c \right ) y+\left (1-n \right ) x^{2}+\left (2 n -1\right ) x -n \right ) y^{\prime }&=2 a y^{2}+2 y x \\ \end{align*}

Unknown ode type.

2.25.15.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 482

ode:=((a*x+c)*y(x)+(1-n)*x^2+(2*n-1)*x-n)*diff(y(x),x) = 2*a*y(x)^2+2*y(x)*x; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-2 \left (a +c \right ) \left (\operatorname {LegendreQ}\left (-\frac {2 n +1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right ) c_1 -\operatorname {LegendreP}\left (\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )\right ) n \sqrt {\frac {y a^{2}}{n \left (a +c \right )}}+\left (\operatorname {LegendreQ}\left (-\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right ) c_1 -\operatorname {LegendreP}\left (-\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )\right ) \left (a +c +\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}\right )}{-2 n \sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right ) \operatorname {LegendreQ}\left (-\frac {2 n +1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )+\left (a +c +\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}\right ) \operatorname {LegendreQ}\left (-\frac {1}{2 n}, \frac {\sqrt {-\left (a +c \right ) \left (4 a n -a -c \right )}}{2 \left (a +c \right ) n}, -\frac {a \left (x -1\right )}{\sqrt {\frac {y a^{2}}{n \left (a +c \right )}}\, \left (a +c \right )}\right )} = 0 \]

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 
Looking for potential symmetries 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\left (a x +c \right ) y \left (x \right )-13629 x^{2}+27259 x -13630\right ) \left (\frac {d}{d x}y \left (x \right )\right )=2 a y \left (x \right )^{2}+2 x y \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {2 a y \left (x \right )^{2}+2 x y \left (x \right )}{\left (a x +c \right ) y \left (x \right )-13629 x^{2}+27259 x -13630} \end {array} \]

2.25.15.2 ✗ 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

2.25.15.3 ✗ 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