55.19.26 problem 26

Internal problem ID [13511]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 26
Date solved : Sunday, October 12, 2025 at 03:36:00 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

\[ y = -a \ln \left (x \right )+\frac {1}{c_1 -\int \frac {f \left (x \right )}{x}d x} \]

✓ Mathematica. Time used: 0.193 (sec). Leaf size: 42

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

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

✗ Sympy

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

IndexError : Index out of range: a[1]