61.8.8 problem 17

Internal problem ID [12089]
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 : 17
Date solved : Sunday, March 30, 2025 at 10:39:56 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 32

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

\[ y = \frac {-b \ln \left (x \right ) a +\tan \left (\left (\ln \left (x \right )+c_1 \right ) \sqrt {a b}\right ) \sqrt {a b}}{a^{2}} \]

✓ Mathematica. Time used: 4.308 (sec). Leaf size: 43

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

\[ y(x)\to -\frac {b \log (x)}{a}+\sqrt {\frac {b}{a^3}} \tan \left (a^2 \sqrt {\frac {b}{a^3}} \log (x)+c_1\right ) \]

✓ Sympy. Time used: 15.326 (sec). Leaf size: 116

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

\[ y{\left (x \right )} = \frac {b \sqrt {- \frac {1}{a^{3} b}} e^{\frac {2 \left (C_{1} + \log {\left (x \right )}\right )}{a \sqrt {- \frac {1}{a^{3} b}}}} + b \sqrt {- \frac {1}{a^{3} b}} - e^{\frac {2 \left (C_{1} + \log {\left (x \right )}\right )}{a \sqrt {- \frac {1}{a^{3} b}}}} \log {\left (x^{\frac {b}{a}} \right )} + \log {\left (x^{\frac {b}{a}} \right )}}{e^{\frac {2 \left (C_{1} + \log {\left (x \right )}\right )}{a \sqrt {- \frac {1}{a^{3} b}}}} - 1} \]