61.11.4 problem 30

Internal problem ID [12204]
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
Date solved : Tuesday, January 28, 2025 at 01:07:19 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=a y^{2}+2 a b \tan \left (x \right ) y+b \left (a b -1\right ) \tan \left (x \right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 81

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)
 

\[ y = \frac {2 c_{1} a b -2 i a^{{3}/{2}} b^{{3}/{2}} \tan \left (x \right ) c_{1} +i \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}-{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x} \tan \left (x \right ) a b}{a \left (2 i c_{1} \sqrt {a}\, \sqrt {b}+{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}\right )} \]

✓ Solution by Mathematica

Time used: 7.716 (sec). Leaf size: 37

DSolve[D[y[x],x]==a*y[x]^2+2*a*b*Tan[x]*y[x]+b*(a*b-1)*Tan[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -b \tan (x)+\sqrt {\frac {b}{a}} \tan \left (a x \sqrt {\frac {b}{a}}+c_1\right ) \]