Internal problem ID [9739]
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]
\[ \boxed {y^{\prime }-y^{2}-a x \ln \left (b x \right )^{m} y-a \ln \left (b x \right )^{m}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 59
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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int \frac {a \ln \left (x b \right )^{m} x^{2}-2}{x}d x}}{c_{1} -\left (\int {\mathrm e}^{\int \frac {a \ln \left (x b \right )^{m} x^{2}-2}{x}d x}d x \right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 2.007 (sec). Leaf size: 93
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+\frac {\exp \left (-\frac {a \log ^{m+1}(b x) \operatorname {ExpIntegralE}(-m,-2 \log (b x))}{b^2}\right )}{\int _1^x\frac {\exp \left (-\frac {a \operatorname {ExpIntegralE}(-m,-2 \log (b K[1])) \log ^{m+1}(b K[1])}{b^2}\right )}{K[1]^2}dK[1]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}