61.11.11 problem 37
Internal
problem
ID
[12132]
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
:
37
Date
solved
:
Friday, March 14, 2025 at 04:13:50 AM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \tan \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right ) \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 351
ode:=(a*tan(lambda*x)+b)*diff(y(x),x) = y(x)^2+k*tan(x*mu)*y(x)-d^2+k*d*tan(x*mu);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\left (\sec \left (\lambda x \right )^{2}\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (a \tan \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda k \left (a^{2}+b^{2}\right ) \left (\int \frac {\tan \left (\mu x \right )}{a \tan \left (\lambda x \right )+b}d x \right )-2 \arctan \left (\tan \left (\lambda x \right )\right ) b d}{\lambda \left (a^{2}+b^{2}\right )}}-d \left (\int \left (a \tan \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda k \left (a^{2}+b^{2}\right ) \left (\int \frac {\tan \left (\mu x \right )}{a \tan \left (\lambda x \right )+b}d x \right )-2 \arctan \left (\tan \left (\lambda x \right )\right ) b d}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_{1} \right )}{\int \left (a \tan \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda k \left (a^{2}+b^{2}\right ) \left (\int \frac {\tan \left (\mu x \right )}{a \tan \left (\lambda x \right )+b}d x \right )-2 \arctan \left (\tan \left (\lambda x \right )\right ) b d}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_{1}}
\]
✓ Mathematica. Time used: 26.191 (sec). Leaf size: 800
ode=(a*Tan[\[Lambda]*x]+b)*D[y[x],x]==y[x]^2+k*Tan[\[Mu]*x]*y[x]-d^2+k*d*Tan[\[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")
d = symbols("d")
k = symbols("k")
cg = symbols("cg")
mu = symbols("mu")
y = Function("y")
ode = Eq(d**2 - d*k*tan(mu*x) - k*y(x)*tan(mu*x) + (a*tan(cg*x) + b)*Derivative(y(x), x) - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : Invalid NaN comparison