61.12.9 problem 46
Internal
problem
ID
[12141]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.6-4.
Equations
with
cotangent.
Problem
number
:
46
Date
solved
:
Sunday, March 30, 2025 at 11:14:07 PM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 366
ode:=(a*cot(lambda*x)+b)*diff(y(x),x) = y(x)^2+c*cot(x*mu)*y(x)-d^2+c*d*cot(x*mu);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-{\mathrm e}^{\frac {c \int \frac {\cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x \lambda \left (a^{2}+b^{2}\right )-2 b \left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}\right ) d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (\lambda x \right )^{2}\right )^{-\frac {d a}{\lambda \left (a^{2}+b^{2}\right )}} \left (a \cot \left (\lambda x \right )+b \right )^{\frac {2 d a}{\lambda \left (a^{2}+b^{2}\right )}}-\left (\int \left (a \cot \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {c \int \frac {\cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x \lambda \left (a^{2}+b^{2}\right )-2 b \left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}\right ) d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (\lambda x \right )^{2}\right )^{-\frac {d a}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_1 \right ) d}{\int \left (a \cot \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {c \int \frac {\cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x \lambda \left (a^{2}+b^{2}\right )-2 b \left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}\right ) d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (\lambda x \right )^{2}\right )^{-\frac {d a}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_1}
\]
✓ Mathematica. Time used: 35.105 (sec). Leaf size: 799
ode=(a*Cot[\[Lambda]*x]+b)*D[y[x],x]==y[x]^2+c*Cot[\[Mu]*x]*y[x]-d^2+c*d*Cot[\[Mu]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
lambda_ = symbols("lambda_")
mu = symbols("mu")
y = Function("y")
ode = Eq(-c*d/tan(mu*x) - c*y(x)/tan(mu*x) + d**2 + (a/tan(lambda_*x) + b)*Derivative(y(x), x) - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : Invalid NaN comparison