55.5.6 problem 6
Internal
problem
ID
[13350]
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
:
6
Date
solved
:
Wednesday, October 01, 2025 at 08:04:37 AM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right ) \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 253
ode:=(sinh(lambda*x)*a+b)*diff(y(x),x) = y(x)^2+c*sinh(x*mu)*y(x)-d^2+c*d*sinh(x*mu);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-d \int \frac {{\mathrm e}^{\frac {c \int \frac {\sinh \left (\mu x \right )}{a \sinh \left (\lambda x \right )+b}d x \sqrt {a^{2}+b^{2}}\, \lambda +4 d \,\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{a \sinh \left (\lambda x \right )+b}d x +d c_1 -{\mathrm e}^{\frac {c \int \frac {\sinh \left (\mu x \right )}{a \sinh \left (\lambda x \right )+b}d x \sqrt {a^{2}+b^{2}}\, \lambda +4 d \,\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\int \frac {{\mathrm e}^{\frac {c \int \frac {\sinh \left (\mu x \right )}{a \sinh \left (\lambda x \right )+b}d x \sqrt {a^{2}+b^{2}}\, \lambda +4 d \,\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{a \sinh \left (\lambda x \right )+b}d x -c_1}
\]
✓ Mathematica. Time used: 4.033 (sec). Leaf size: 289
ode=(a*Sinh[\[Lambda]*x]+b)*D[y[x],x]==y[x]^2+c*Sinh[\[Mu]*x]*y[x]-d^2+c*d*Sinh[\[Mu]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right ) (-d+c \sinh (\mu K[2])+y(x))}{c \mu (b+a \sinh (\lambda K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right ) (-d+K[3]+c \sinh (\mu K[2]))}{c \mu (d+K[3])^2 (b+a \sinh (\lambda K[2]))}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \sinh (\mu K[1])}{b+a \sinh (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3]) (b+a \sinh (\lambda K[2]))}\right )dK[2]\right )dK[3]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
lambda_ = symbols("lambda_")
mu = symbols("mu")
y = Function("y")
ode = Eq(-c*d*sinh(mu*x) - c*y(x)*sinh(mu*x) + d**2 + (a*sinh(lambda_*x) + b)*Derivative(y(x), x) - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out