61.8.5 problem 14

Internal problem ID [12086]
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 : 14
Date solved : Wednesday, March 05, 2025 at 04:14:51 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 51

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

\[ y = \left (b \left (\int {\mathrm e}^{a \left (\int \ln \left (x \right )^{n} \left (-1+\ln \left (x \right ) b x \right )d x \right )} \left (1+\ln \left (x \right )\right )d x \right )+c_{1} \right ) {\mathrm e}^{-a \left (\int \ln \left (x \right )^{n} \left (-1+\ln \left (x \right ) b x \right )d x \right )} \]

✓ Mathematica. Time used: 0.225 (sec). Leaf size: 78

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

\[ y(x)\to \exp \left (\int _1^x-a \log ^n(K[1]) (b K[1] \log (K[1])-1)dK[1]\right ) \left (\int _1^xb \exp \left (-\int _1^{K[2]}-a \log ^n(K[1]) (b K[1] \log (K[1])-1)dK[1]\right ) (\log (K[2])+1)dK[2]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*b*x*y(x)*log(x)**(n + 1) - a*y(x)*log(x)**n - b*log(x) - b + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out