55.11.10 problem 36

Internal problem ID [13430]
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 : 36
Date solved : Wednesday, October 01, 2025 at 12:00:04 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 31

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

\[ y = -\tan \left (-b a \int \tan \left (\lambda x \right )^{m} x^{k -1}d x +c_1 \right ) b \,x^{k} \]

✓ Mathematica. Time used: 0.452 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^xa K[1]^{k-1} \tan ^m(\lambda K[1])dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

Timed Out