61.13.7 problem 53

Internal problem ID [12148]
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
Date solved : Wednesday, March 05, 2025 at 04:55:24 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=diff(y(x),x) = y(x)^2-y(x)*tan(x)+a*(-a+1)*cot(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\cot \left (x \right ) \sin \left (x \right )^{2 a} a +c_{1} \cos \left (x \right ) \left (a -1\right )}{\sin \left (x \right ) c_{1} +\sin \left (x \right )^{2 a}} \]

✓ Mathematica. Time used: 5.182 (sec). Leaf size: 230

ode=D[y[x],x]==y[x]^2-y[x]*Tan[x]+a*(1-a)*Cot[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {i \cot (x) \left (\left (\sqrt {a-1} \sqrt {a} \sqrt {-\frac {(2 a-1)^2}{(a-1) a}}-i\right ) \left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {a-1} \sqrt {a} \sqrt {-\frac {(2 a-1)^2}{(a-1) a}}}-\left (\sqrt {a-1} \sqrt {a} \sqrt {-\frac {(2 a-1)^2}{(a-1) a}}+i\right ) c_1\right )}{2 \left (\left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4}}+c_1\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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(1 - a)/tan(x)**2 - y(x)**2 + y(x)*tan(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-a**2 + a + (y(x) - tan(x))*y(x)*tan(x)**2)/tan(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method