Internal problem ID [9791]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-3. Equations with tangent.
Problem number: 36.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \left (\tan ^{m}\left (\lambda x \right )\right ) y^{2}-k y-a \,b^{2} x^{2 k} \left (\tan ^{m}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 36
dsolve(x*diff(y(x),x)=a*tan(lambda*x)^m*y(x)^2+k*y(x)+a*b^2*x^(2*k)*tan(lambda*x)^m,y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (-b a \left (\int \frac {x^{k} \left (\tan ^{m}\left (\lambda x \right )\right )}{x}d x \right )+c_{1}\right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 1.065 (sec). Leaf size: 50
DSolve[x*y'[x]==a*Tan[\[Lambda]*x]^m*y[x]^2+k*y[x]+a*b^2*x^(2*k)*Tan[\[Lambda]*x]^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^xa K[1]^{k-1} \tan ^m(\lambda K[1])dK[1]+c_1\right ) \\ \end{align*}