61.8.2 problem 11

Internal problem ID [12083]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number : 11
Date solved : Sunday, March 30, 2025 at 10:39:23 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

ode:=diff(y(x),x) = y(x)^2+a*x*ln(b*x)^m*y(x)+a*ln(b*x)^m; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x}}{c_1 -\int {\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x}d x}-\frac {1}{x} \]

✓ Mathematica. Time used: 2.284 (sec). Leaf size: 126

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

\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-a K[1] \log ^m(b K[1])dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \log ^m(b K[1])dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \log ^m(b K[1])dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -a*x*y(x)*log(b*x)**m - a*log(b*x)**m - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method