61.5.12 problem 12
Internal
problem
ID
[12057]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.4-1.
Equations
with
hyperbolic
sine
and
cosine
Problem
number
:
12
Date
solved
:
Wednesday, March 05, 2025 at 03:59:38 PM
CAS
classification
:
[_Riccati]
\begin{align*} 2 y^{\prime }&=\left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right ) \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 101
ode:=2*diff(y(x),x) = (a-lambda+a*cosh(lambda*x))*y(x)^2+a+lambda-a*cosh(lambda*x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\tanh \left (\frac {\lambda x}{2}\right ) \lambda \left (\int {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2} {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }} c_{1} \lambda -2 \tanh \left (\frac {\lambda x}{2}\right )}{\lambda \left (\int {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) c_{1} -2}
\]
✓ Mathematica. Time used: 16.686 (sec). Leaf size: 338
ode=2*D[y[x],x]==(a-\[Lambda]+a*Cosh[\[Lambda]*x])*y[x]^2+a+\[Lambda]-a*Cosh[\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\text {sech}^2\left (\frac {\lambda x}{2}\right ) \left (c_1 \sinh (\lambda x) \int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]+4 c_1 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\sinh (\lambda x)\right )}{2+2 c_1 \int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]} \\
y(x)\to \frac {1}{2} \text {sech}^2\left (\frac {\lambda x}{2}\right ) \left (\frac {4 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]}+\sinh (\lambda x)\right ) \\
y(x)\to \frac {1}{2} \text {sech}^2\left (\frac {\lambda x}{2}\right ) \left (\frac {4 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]}+\sinh (\lambda x)\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
cg = symbols("cg")
y = Function("y")
ode = Eq(a*cosh(cg*x) - a - cg - (a*cosh(cg*x) + a - cg)*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*cosh(cg*x)/2 - a*y(x)**2/2 + a*cosh(cg*x)/2 - a/2 + cg*y(x)**2/2 - cg/2 + Derivative(y(x), x) cannot be solved by the factorable group method