61.7.2 problem 2

Internal problem ID [12153]
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 : Tuesday, January 28, 2025 at 12:44:54 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[x*D[y[x],x]==a*y[x]^2+b*Log[x]+c,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*}