55.2.73 problem 73

Internal problem ID [13299]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 73
Date solved : Wednesday, October 01, 2025 at 06:23:29 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 138

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

\[ y = -\frac {\left (\tanh \left (\frac {\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 \alpha \gamma +\beta ^{2}\right )}\, \left (\left (-b n -\beta \right ) \ln \left (a \,x^{k}+b \right )+k \left (\left (b n +\beta \right ) \ln \left (x \right )+b c_1 \right )\right )}{2 b k \left (b n +\beta \right )^{2}}\right ) \sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 \alpha \gamma +\beta ^{2}\right )}+\left (b n +\beta \right )^{2}\right ) x^{-n}}{2 \alpha \left (b n +\beta \right )} \]

✓ Mathematica. Time used: 2.137 (sec). Leaf size: 663

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

Timed Out