55.11.7 problem 33

Internal problem ID [13427]
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 : 33
Date solved : Wednesday, October 01, 2025 at 11:43:49 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 174

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

\[ y = \frac {x^{-k -1} \left (x^{k +1} {\mathrm e}^{\int \frac {\tan \left (x \right )^{m} x^{k +1} a x -2 k -2}{x}d x}+\int x^{k} {\mathrm e}^{\int \frac {\tan \left (x \right )^{m} x^{k +1} a x -2 k -2}{x}d x}d x k +\int x^{k} {\mathrm e}^{\int \frac {\tan \left (x \right )^{m} x^{k +1} a x -2 k -2}{x}d x}d x -c_1 \right )}{\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \tan \left (x \right )^{m}-2 k -2}{x}d x}d x k +\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \tan \left (x \right )^{m}-2 k -2}{x}d x}d x -c_1} \]

✓ Mathematica. Time used: 2.769 (sec). Leaf size: 248

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

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

✗ Sympy

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

Timed Out