61.10.7 problem 20
Internal
problem
ID
[12115]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.6-2.
Equations
with
cosine.
Problem
number
:
20
Date
solved
:
Sunday, March 30, 2025 at 10:54:16 PM
CAS
classification
:
[_Riccati]
\begin{align*} 2 y^{\prime }&=\left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right ) \end{align*}
✓ Maple. Time used: 0.071 (sec). Leaf size: 120
ode:=2*diff(y(x),x) = (lambda+a-a*cos(lambda*x))*y(x)^2+lambda-a-a*cos(lambda*x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\lambda \int {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) \left (\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}+2 a \right )d x c_1 \cot \left (\frac {\lambda x}{2}\right )-2 \,{\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) c_1 \lambda \csc \left (\frac {\lambda x}{2}\right )^{2}+2 i \cot \left (\frac {\lambda x}{2}\right )}{\lambda \int {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) \left (\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}+2 a \right )d x c_1 -2 i}
\]
✓ Mathematica. Time used: 10.395 (sec). Leaf size: 321
ode=2*D[y[x],x]==(\[Lambda]+a-a*Cos[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Cos[\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {2 \left (c_1 \cot \left (\frac {\lambda x}{2}\right ) \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]+2 c_1 \csc ^2\left (\frac {\lambda x}{2}\right ) e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\cot \left (\frac {\lambda x}{2}\right )\right )}{2+2 c_1 \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]} \\
y(x)\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \\
y(x)\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(a*cos(lambda_*x) + a - lambda_ - (-a*cos(lambda_*x) + a + lambda_)*y(x)**2 + 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a*y(x)**2*cos(lambda_*x)/2 - a*y(x)**2/2 + a*cos(lambda_*x)/2 + a/2 - lambda_*y(x)**2/2 - lambda_/2 + Derivative(y(x), x) cannot be solved by the factorable group method