Internal problem ID [9750]
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: 18.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }-a \ln \left (\lambda x \right )^{m} y^{2}-k y-a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 36
dsolve(x*diff(y(x),x)=a*(ln(lambda*x))^m*y(x)^2+k*y(x)+a*b^2*x^(2*k)*(ln(lambda*x))^m,y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (-a b \left (\int \frac {\ln \left (\lambda x \right )^{m} x^{k}}{x}d x \right )+c_{1}\right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 1.498 (sec). Leaf size: 59
DSolve[x*y'[x]==a*(Log[\[Lambda]*x])^m*y[x]^2+k*y[x]+a*b^2*x^(2*k)*(Log[\[Lambda]*x])^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {b^2} x^k \tan \left (-a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^{m+1}(\lambda x) E_{-m}(-k \log (x \lambda ))+c_1\right ) \\ \end{align*}