61.9.2 problem 2
Internal
problem
ID
[12097]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.6-1.
Equations
with
sine
Problem
number
:
2
Date
solved
:
Sunday, March 30, 2025 at 10:41:29 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.071 (sec). Leaf size: 289
ode:=diff(y(x),x) = y(x)^2-a^2+a*lambda*sin(lambda*x)+a^2*sin(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-2 a c_1 \cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )-c_1 \lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )-2 \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \left (\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_1 \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{2}\right ) \cos \left (\lambda x \right )}{2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) c_1 +2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}
\]
✓ Mathematica. Time used: 1.766 (sec). Leaf size: 157
ode=D[y[x],x]==y[x]^2-a^2+a*\[Lambda]*Sin[\[Lambda]*x]+a^2*Sin[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {c_1 \exp \left (\int _1^x-2 a \cos (\lambda K[1])dK[1]\right )+a c_1 \cos (\lambda x) \int _1^x\exp \left (\int _1^{K[2]}-2 a \cos (\lambda K[1])dK[1]\right )dK[2]+a \cos (\lambda x)}{1+c_1 \int _1^x\exp \left (\int _1^{K[2]}-2 a \cos (\lambda K[1])dK[1]\right )dK[2]} \\
y(x)\to -\frac {\exp \left (\int _1^x-2 a \cos (\lambda K[1])dK[1]\right )}{\int _1^x\exp \left (\int _1^{K[2]}-2 a \cos (\lambda K[1])dK[1]\right )dK[2]}-a \cos (\lambda x) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-a**2*sin(lambda_*x)**2 + a**2 - a*lambda_*sin(lambda_*x) - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2*cos(lambda_*x)**2 - a*lambda_*sin(lambda_*x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method