61.5.1 problem 1
Internal
problem
ID
[12046]
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
:
1
Date
solved
:
Wednesday, March 05, 2025 at 03:55:25 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 289
ode:=diff(y(x),x) = y(x)^2-a^2+a*lambda*sinh(lambda*x)-a^2*sinh(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-2 c_{1} a \cosh \left (\lambda x \right ) \sinh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right )-c_{1} \lambda \cosh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\left (-2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) a +i \left (\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{1} \sinh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right ) \lambda \right ) \cosh \left (\lambda x \right )}{2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sinh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )}
\]
✓ Mathematica. Time used: 6.242 (sec). Leaf size: 212
ode=D[y[x],x]==y[x]^2-a^2+a*\[Lambda]*Sinh[\[Lambda]*x]-a^2*Sinh[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {e^{\lambda (-x)} \left (2 \lambda \exp \left (2 \lambda x-2 \int _1^{e^{x \lambda }}-\frac {a K[1]^2-\lambda K[1]+a}{2 \lambda K[1]^2}dK[1]\right )-a \left (e^{2 \lambda x}+1\right ) \int _1^{e^{x \lambda }}\exp \left (-2 \int _1^{K[2]}-\frac {a K[1]^2-\lambda K[1]+a}{2 \lambda K[1]^2}dK[1]\right )dK[2]-a c_1 e^{2 \lambda x}-a c_1\right )}{2 \left (\int _1^{e^{x \lambda }}\exp \left (-2 \int _1^{K[2]}-\frac {a K[1]^2-\lambda K[1]+a}{2 \lambda K[1]^2}dK[1]\right )dK[2]+c_1\right )} \\
y(x)\to \frac {1}{2} a e^{\lambda (-x)} \left (e^{2 \lambda x}+1\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
cg = symbols("cg")
y = Function("y")
ode = Eq(a**2*sinh(cg*x)**2 + a**2 - a*cg*sinh(cg*x) - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2*cosh(cg*x)**2 - a*cg*sinh(cg*x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method