Internal problem ID [10486]
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: 11.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a x \ln \left (b x \right )^{m} y=a \ln \left (b x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 85
dsolve(diff(y(x),x)=y(x)^2+a*x*(ln(b*x))^m*y(x)+a*(ln(b*x))^m,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]
✓ Solution by Mathematica
Time used: 3.589 (sec). Leaf size: 181
DSolve[y'[x]==y[x]^2+a*x*(Log[b*x])^m*y[x]+a*(Log[b*x])^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \int _1^x\frac {\exp \left (\frac {2^{-m-1} a \Gamma (m+1,-2 \log (b K[1])) (-\log (b K[1]))^{-m} \log ^m(b K[1])}{b^2}\right )}{K[1]^2}dK[1]+\exp \left (\frac {a 2^{-m-1} (-\log (b x))^{-m} \log ^m(b x) \Gamma (m+1,-2 \log (b x))}{b^2}\right )+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (\frac {2^{-m-1} a \Gamma (m+1,-2 \log (b K[1])) (-\log (b K[1]))^{-m} \log ^m(b K[1])}{b^2}\right )}{K[1]^2}dK[1]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}