61.10.1 problem 14
Internal
problem
ID
[12109]
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
:
14
Date
solved
:
Sunday, March 30, 2025 at 10:50:39 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right ) \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 94
ode:=diff(y(x),x) = alpha*y(x)^2+beta+gamma*cos(lambda*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\lambda \left (c_1 \operatorname {MathieuSPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, \frac {\lambda x}{2}\right )+\operatorname {MathieuCPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, \frac {\lambda x}{2}\right )\right )}{2 \alpha \left (c_1 \operatorname {MathieuS}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, \frac {\lambda x}{2}\right )+\operatorname {MathieuC}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, \frac {\lambda x}{2}\right )\right )}
\]
✓ Mathematica. Time used: 0.261 (sec). Leaf size: 163
ode=D[y[x],x]==\[Alpha]*y[x]^2+\[Beta]+\[Gamma]*Cos[\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\lambda \left (\text {MathieuSPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]+c_1 \text {MathieuCPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]\right )}{2 \alpha \left (\text {MathieuS}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]+c_1 \text {MathieuC}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]\right )} \\
y(x)\to -\frac {\lambda \text {MathieuCPrime}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]}{2 \alpha \text {MathieuC}\left [\frac {4 \alpha \beta }{\lambda ^2},-\frac {2 \alpha \gamma }{\lambda ^2},\frac {\lambda x}{2}\right ]} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
Gamma = symbols("Gamma")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-Alpha*y(x)**2 - BETA - Gamma*cos(lambda_*x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -Alpha*y(x)**2 - BETA - Gamma*cos(lambda_*x) + Derivative(y(x), x) cannot be solved by the lie group method