2.25.6 Problem 8

Internal problem ID [13621]
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 : 8
Date solved : Friday, December 19, 2025 at 08:55:14 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.25.6.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 4278

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

\[ \text {Expression too large to display} \]

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=-13620 y \left (x \right )^{2}+\left (27243 a x +27241\right ) y \left (x \right )-13621 a^{2} x^{2}-b x -13621 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-13620 y \left (x \right )^{2}+\left (27243 a x +27241\right ) y \left (x \right )-13621 a^{2} x^{2}-b x -13621}{2 x y \left (x \right )} \end {array} \]

2.25.6.2 ✗ Mathematica

ode=2*x*y[x]*D[y[x],x]==(1-n)*y[x]^2+(a*(2*n+1)*x+2*n-1)*y[x]-a^2*n*x^2-b*x-n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.25.6.3 ✗ 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 + b*x + n + 2*x*y(x)*Derivative(y(x), x) - (1 - n)*y(x)**2 - (a*x*(2*n + 1) + 2*n - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out