61.13.11 problem 57

Internal problem ID [12152]
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 : 57
Date solved : Sunday, March 30, 2025 at 11:21:18 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 268

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

\[ y = \frac {4 c_1 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {-a -b +2 \lambda }{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 c_1 \left (\left (-3 \lambda ^{2}+\left (\frac {7 a}{2}+\frac {3 b}{2}\right ) \lambda -a b \right ) \cos \left (\lambda x \right )^{2}+a^{2} \sin \left (\lambda x \right )^{2}-\frac {5 \left (a -\frac {3 \lambda }{5}\right ) \lambda }{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right ) \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} \left (-\frac {3 \lambda }{2}+a \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_1 \cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}\right )} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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