Internal problem ID [9781]
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: 30.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime }-y^{2} a -2 a b \tan \left (x \right ) y-b \left (a b -1\right ) \tan \left (x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 60
dsolve(diff(y(x),x)=a*y(x)^2+2*a*b*tan(x)*y(x)+b*(a*b-1)*tan(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-b \tan \left (x \right ) a -i \sqrt {b}\, \sqrt {a}+\frac {{\mathrm e}^{-2 i \sqrt {b}\, \sqrt {a}\, x}}{c_{1} -\frac {i {\mathrm e}^{-2 i \sqrt {b}\, \sqrt {a}\, x}}{2 \sqrt {b}\, \sqrt {a}}}}{a} \]
✓ Solution by Mathematica
Time used: 8.415 (sec). Leaf size: 37
DSolve[y'[x]==a*y[x]^2+2*a*b*Tan[x]*y[x]+b*(a*b-1)*Tan[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -b \tan (x)+\sqrt {\frac {b}{a}} \tan \left (a x \sqrt {\frac {b}{a}}+c_1\right ) \\ \end{align*}