61.8.9 problem 18

Internal problem ID [12090]
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 : 18
Date solved : Wednesday, March 05, 2025 at 04:15:09 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 31

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

\[ y = -\tan \left (-a b \left (\int x^{k -1} \ln \left (\lambda x \right )^{m}d x \right )+c_{1} \right ) b \,x^{k} \]

✓ Mathematica. Time used: 1.345 (sec). Leaf size: 70

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

\[ y(x)\to \sqrt {b^2} x^k \tan \left (\frac {a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^m(\lambda x) (-k \log (\lambda x))^{-m} \Gamma (m+1,-k \log (x \lambda ))}{k}+c_1\right ) \]

✗ Sympy

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

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