61.8.1 problem 10

Internal problem ID [12082]
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 : 10
Date solved : Sunday, March 30, 2025 at 10:39:20 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 48

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

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

✓ Mathematica. Time used: 0.602 (sec). Leaf size: 187

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

\[ \text {Solve}\left [\int _1^x\frac {e^{2 b K[1]-a K[1]} (\beta K[1])^{a K[1]} (b+a \log (\beta K[1])+y(x))}{a (b-y(x))}dK[1]+\int _1^{y(x)}\left (\frac {e^{2 b x-a x} (x \beta )^{a x}}{a (K[2]-b)^2}-\int _1^x\left (\frac {e^{2 b K[1]-a K[1]} (b+K[2]+a \log (\beta K[1])) (\beta K[1])^{a K[1]}}{a (b-K[2])^2}+\frac {e^{2 b K[1]-a K[1]} (\beta K[1])^{a K[1]}}{a (b-K[2])}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 5.096 (sec). Leaf size: 85

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

\[ y{\left (x \right )} = \frac {\left (C_{1} b e^{a x} - b e^{a x} \int e^{- a x} e^{2 b x} e^{a x \log {\left (\beta x \right )}}\, dx + e^{x \left (a \log {\left (\beta x \right )} + 2 b\right )}\right ) e^{- a x}}{C_{1} - \int e^{- a x} e^{2 b x} e^{a x \log {\left (\beta x \right )}}\, dx} \]