Internal problem ID [9782]
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: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a \tan \left (\beta x \right ) y-a b \tan \left (\beta x \right )+b^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(diff(y(x),x)=y(x)^2+a*tan(beta*x)*y(x)+a*b*tan(beta*x)-b^2,y(x), singsol=all)
\[ y \left (x \right ) = -b -\frac {\left (1+\tan \left (\beta x \right )^{2}\right )^{\frac {a}{2 \beta }} {\mathrm e}^{-2 x b}}{\int \left (1+\tan \left (\beta x \right )^{2}\right )^{\frac {a}{2 \beta }} {\mathrm e}^{-2 x b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 16.405 (sec). Leaf size: 329
DSolve[y'[x]==y[x]^2+a*Tan[\[Beta]*x]*y[x]+a*b*Tan[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{-\frac {a}{\beta }} (a+2 i b+2 \beta ) \cos ^{-\frac {a}{\beta }}(\beta x) \left (b \left (2 i a \operatorname {Hypergeometric2F1}\left (1,-\frac {a-2 i b}{2 \beta },\frac {a+2 i b}{2 \beta }+1,-e^{2 i x \beta }\right )-i a+2 b\right ) (\sin (2 \beta x) \csc (\beta x))^{a/\beta }-2^{a/\beta } \left (a^2+4 b^2\right ) \cos ^{\frac {a}{\beta }}(\beta x) \left (1+a b \beta c_1 e^{2 b x} \cos ^{\frac {a}{\beta }}(\beta x)\right )\right )}{(a-2 i b) \left ((2 b-i a) e^{2 i \beta x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {a-2 i b}{2 \beta },\frac {a+2 i b}{2 \beta }+2,-e^{2 i x \beta }\right )+(a+2 i b+2 \beta ) \left (a \beta c_1 (a+2 i b) e^{2 b x} \cos ^{\frac {a}{\beta }}(\beta x)-i \operatorname {Hypergeometric2F1}\left (1,-\frac {a-2 i b}{2 \beta },\frac {a+2 i b}{2 \beta }+1,-e^{2 i x \beta }\right )\right )\right )} \\ y(x)\to -b \\ \end{align*}