61.11.1 problem 27
Internal
problem
ID
[12122]
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
:
27
Date
solved
:
Sunday, March 30, 2025 at 10:59:40 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.061 (sec). Leaf size: 205
ode:=diff(y(x),x) = y(x)^2+a*lambda+a*(lambda-a)*tan(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\sec \left (\lambda x \right ) \left (\sin \left (\lambda x \right ) \operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) a +\sin \left (\lambda x \right ) \operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_1 a -\lambda \left (c_1 \operatorname {LegendreQ}\left (\frac {\lambda +2 a}{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {\lambda +2 a}{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )\right )\right )}{\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_1 +\operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )}
\]
✓ Mathematica. Time used: 1.671 (sec). Leaf size: 259
ode=D[y[x],x]==y[x]^2+a*\[Lambda]+a*(\[Lambda]-a)*Tan[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2 \left (a c_1 \sin ^2(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )+(2 a-\lambda ) \sqrt {\sin ^2(\lambda x)} \left (a \sin (\lambda x) \cos ^{\frac {2 a}{\lambda }-1}(\lambda x)-c_1\right )\right )}{2 (2 a-\lambda ) \sqrt {\sin ^2(\lambda x)} \cos ^{\frac {2 a}{\lambda }}(\lambda x)+c_1 \sin (2 \lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )} \\
y(x)\to \frac {\tan (\lambda x) \left (a \sqrt {\sin ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )-2 a+\lambda \right )}{\sqrt {\sin ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-a*lambda_ - a*(-a + 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 - a*lambda_*tan(lambda_*x)**2 - a*lambda_ - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method