Internal problem ID [10494]
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: 9.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )=1} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve(x^2*ln(a*x)*(diff(y(x),x)-y(x)^2)=1,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {c_{1} \operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )-1}{x \left (\left (c_{1} \operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )-1\right ) \ln \left (a x \right )+a x c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 0.616 (sec). Leaf size: 74
DSolve[x^2*Log[a*x]*(y'[x]-y[x]^2)==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a+c_1 \operatorname {LogIntegral}(a x)}{-c_1 x \operatorname {LogIntegral}(a x) \log (a x)+a c_1 x^2-a x \log (a x)} y(x)\to \frac {\operatorname {LogIntegral}(a x)}{a x^2-x \operatorname {LogIntegral}(a x) \log (a x)} \end{align*}