61.7.2 problem 2

Internal problem ID [12074]
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 : 2
Date solved : Wednesday, March 05, 2025 at 04:13:26 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 91

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

\[ y = \frac {\left (a b \right )^{{1}/{3}} \left (\operatorname {AiryBi}\left (1, -\frac {\left (a b \right )^{{1}/{3}} \left (\ln \left (x \right ) b +c \right )}{b}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {\left (a b \right )^{{1}/{3}} \left (\ln \left (x \right ) b +c \right )}{b}\right )\right )}{a \left (c_{1} \operatorname {AiryBi}\left (-\frac {\left (a b \right )^{{1}/{3}} \left (\ln \left (x \right ) b +c \right )}{b}\right )+\operatorname {AiryAi}\left (-\frac {\left (a b \right )^{{1}/{3}} \left (\ln \left (x \right ) b +c \right )}{b}\right )\right )} \]

✓ Mathematica. Time used: 0.803 (sec). Leaf size: 149

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

\begin{align*} y(x)\to \frac {b \left (\operatorname {AiryBiPrime}\left (-\frac {a (c+b \log (x))}{(-a b)^{2/3}}\right )+c_1 \operatorname {AiryAiPrime}\left (-\frac {a (c+b \log (x))}{(-a b)^{2/3}}\right )\right )}{(-a b)^{2/3} \left (\operatorname {AiryBi}\left (-\frac {a (c+b \log (x))}{(-a b)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (-\frac {a (c+b \log (x))}{(-a b)^{2/3}}\right )\right )} \\ y(x)\to \frac {b \operatorname {AiryAiPrime}\left (-\frac {a (c+b \log (x))}{(-a b)^{2/3}}\right )}{(-a b)^{2/3} \operatorname {AiryAi}\left (-\frac {a (c+b \log (x))}{(-a b)^{2/3}}\right )} \\ \end{align*}

✗ Sympy

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

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