61.7.9 problem 9

Internal problem ID [12081]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-1. Equations Containing Logarithmic Functions
Problem number : 9
Date solved : Sunday, March 30, 2025 at 10:39:17 PM
CAS classification : [_Riccati]

\begin{align*} x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )&=1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

ode:=x^2*ln(a*x)*(diff(y(x),x)-y(x)^2) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-c_1 \,\operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )-1}{x \left (\left (c_1 \,\operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )+1\right ) \ln \left (a x \right )+c_1 a x \right )} \]

✓ Mathematica. Time used: 0.754 (sec). Leaf size: 102

ode=x^2*Log[a*x]*(D[y[x],x]-y[x]^2)==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*(-y(x)**2 + Derivative(y(x), x))*log(a*x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -y(x)**2 + Derivative(y(x), x) - 1/(x**2*log(a*x)) cannot be solved by the factorable group method