61.6.9 problem 26
Internal
problem
ID
[12071]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.4-2.
Equations
with
hyperbolic
tangent
and
cotangent.
Problem
number
:
26
Date
solved
:
Wednesday, March 05, 2025 at 04:12:33 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 143
ode:=diff(y(x),x) = y(x)^2-2*lambda^2*tanh(lambda*x)^2-2*lambda^2*coth(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 \lambda \,\operatorname {sech}\left (\lambda x \right ) \left (-\frac {1}{2}+c_{1} \left (-\cosh \left (\lambda x \right )^{2}+\frac {1}{2}\right ) \ln \left (\coth \left (\lambda x \right )-1\right )+\left (\cosh \left (\lambda x \right )^{2}-\frac {1}{2}\right ) c_{1} \ln \left (\coth \left (\lambda x \right )+1\right )+4 \cosh \left (\lambda x \right )^{5} c_{1} \sinh \left (\lambda x \right )-4 \cosh \left (\lambda x \right )^{3} c_{1} \sinh \left (\lambda x \right )-\sinh \left (\lambda x \right ) \cosh \left (\lambda x \right ) c_{1} +\cosh \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )}{-4 \cosh \left (\lambda x \right )^{3} c_{1} \sinh \left (\lambda x \right )+2 \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right ) c_{1} -\ln \left (\coth \left (\lambda x \right )-1\right ) c_{1} +\ln \left (\coth \left (\lambda x \right )+1\right ) c_{1} +1}
\]
✓ Mathematica. Time used: 5.314 (sec). Leaf size: 263
ode=D[y[x],x]==y[x]^2-2*\[Lambda]^2*Tanh[\[Lambda]*x]^2-2*\[Lambda]^2*Coth[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2 \lambda \exp \left (-2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right ) \left (\left (e^{4 \lambda x}+1\right ) \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right ) \int _1^{e^{4 x \lambda }}\exp \left (-2 \int _1^{K[2]}\frac {1}{K[1]-K[1]^2}dK[1]\right )dK[2]+c_1 \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right )+c_1 \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]+4 \lambda x\right )+2 e^{4 \lambda x}-2 e^{8 \lambda x}\right )}{\left (e^{4 \lambda x}-1\right ) \left (\int _1^{e^{4 x \lambda }}\exp \left (-2 \int _1^{K[2]}\frac {1}{K[1]-K[1]^2}dK[1]\right )dK[2]+c_1\right )} \\
y(x)\to \frac {2 \lambda \left (e^{4 \lambda x}+1\right )}{e^{4 \lambda x}-1} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
cg = symbols("cg")
y = Function("y")
ode = Eq(2*cg**2*tanh(cg*x)**2 + 2*cg**2/tanh(cg*x)**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE 2*cg**2*tanh(cg*x)**2 + 2*cg**2/tanh(cg*x)**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method