Internal problem ID [10493]
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]
\[ \boxed {y^{\prime } x -a \ln \left (\lambda x \right )^{m} y^{2}-k y=a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 31
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)
\[ y \left (x \right ) = -\tan \left (-a b \left (\int x^{-1+k} \ln \left (x \lambda \right )^{m}d x \right )+c_{1} \right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 2.161 (sec). Leaf size: 70
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]
\[ y(x)\to \sqrt {b^2} x^k \tan \left (\frac {a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^m(\lambda x) (-k \log (\lambda x))^{-m} \Gamma (m+1,-k \log (x \lambda ))}{k}+c_1\right ) \]