Internal problem ID [10477]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime } x -a y^{2}=b \ln \left (x \right )+c} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 91
dsolve(x*diff(y(x),x)=a*y(x)^2+b*ln(x)+c,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (a b \right )^{\frac {1}{3}} \left (\operatorname {AiryBi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )\right )}{a \left (c_{1} \operatorname {AiryBi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )+\operatorname {AiryAi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.682 (sec). Leaf size: 149
DSolve[x*y'[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*}