Internal problem ID [9804]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 53.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}+\tan \left (x \right ) y-a \left (-a +1\right ) \cot \left (x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 70
dsolve(diff(y(x),x)=y(x)^2-y(x)*tan(x)+a*(1-a)*cot(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {a c_{1} \sin \left (x \right )}{c_{1} \sin \left (x \right )+\sin \left (x \right )^{2 a}}-\frac {c_{1} \sin \left (x \right )}{c_{1} \sin \left (x \right )+\sin \left (x \right )^{2 a}}-\frac {\sin \left (x \right )^{2 a} a}{c_{1} \sin \left (x \right )+\sin \left (x \right )^{2 a}}\right ) \cos \left (x \right )}{\sin \left (x \right )} \]
✓ Solution by Mathematica
Time used: 4.518 (sec). Leaf size: 141
DSolve[y'[x]==y[x]^2-y[x]*Tan[x]+a*(1-a)*Cot[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \cot (x) \left (-1-i \sqrt {\frac {1}{a}+\frac {1}{1-a}-4} \sqrt {a-1} \sqrt {a} \left (1-\frac {2 c_1}{\left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {\frac {1}{a}+\frac {1}{1-a}-4} \sqrt {a-1} \sqrt {a}}+c_1}\right )\right ) \\ y(x)\to \frac {1}{2} i \left (\sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4}+i\right ) \cot (x) \\ \end{align*}