2.25.19 Problem 36

Internal problem ID [13634]
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 : 36
Date solved : Friday, December 19, 2025 at 09:32:06 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.25.19.1 ✓ Maple. Time used: 0.006 (sec). Leaf size: 2162

ode:=(y(x)*x+a*x^n+b*x^2)*diff(y(x),x) = y(x)^2+c*x^n+b*x*y(x); 
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 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x y \left (x \right )+a \,x^{13634}+b \,x^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right )=y \left (x \right )^{2}+c \,x^{13634}+b x y \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )^{2}+c \,x^{13634}+b x y \left (x \right )}{x y \left (x \right )+a \,x^{13634}+b \,x^{2}} \end {array} \]

2.25.19.2 ✓ Mathematica. Time used: 6.05 (sec). Leaf size: 67

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

\[ \text {Solve}\left [-\frac {c \left (\log \left (a^2 (n-1) x^n+a x (b (n-1) x+(n-2) y(x))+c x^2\right )+(n-2) \log (c x-a y(x))-n \log (x)\right )}{n-2}=c_1,y(x)\right ] \]

2.25.19.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
c = symbols("c") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-b*x*y(x) - c*x**n + (a*x**n + b*x**2 + x*y(x))*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out