55.5.7 problem 7
Internal
problem
ID
[13351]
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
:
7
Date
solved
:
Wednesday, October 01, 2025 at 08:06:15 AM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right )&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 250
ode:=(sinh(lambda*x)*a+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*sinh(lambda*x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {4 \lambda \left (a \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (a^{2} b^{2} \operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )+b^{4} \operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )-c_1 \right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+a b \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right )}{\sqrt {a^{2}+b^{2}}\, \left (2 a \cosh \left (\frac {\lambda x}{2}\right ) \left (a^{2}+b^{2}\right )^{{3}/{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (a \sinh \left (\frac {\lambda x}{2}\right ) \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right ) \left (a^{2} b^{2} \operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )+b^{4} \operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )-c_1 \right )\right )}
\]
✓ Mathematica. Time used: 11.48 (sec). Leaf size: 229
ode=(a*Sinh[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)+a*\[Lambda]^2*Sinh[\[Lambda]*x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x-\frac {-a \sinh (\lambda K[1]) \lambda ^2+b y(x)^2+a \sinh (\lambda K[1]) y(x)^2}{(b+a \sinh (\lambda K[1])) (a \lambda \cosh (\lambda K[1])+b y(x)+a \sinh (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(a \lambda \cosh (x \lambda )+b K[2]+a K[2] \sinh (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (-a \sinh (\lambda K[1]) \lambda ^2+b K[2]^2+a K[2]^2 \sinh (\lambda K[1])\right )}{(a \lambda \cosh (\lambda K[1])+b K[2]+a K[2] \sinh (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \sinh (\lambda K[1]) K[2]}{(b+a \sinh (\lambda K[1])) (a \lambda \cosh (\lambda K[1])+b K[2]+a K[2] \sinh (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(a*lambda_**2*sinh(lambda_*x) + (a*sinh(lambda_*x) + b)*(-y(x)**2 + Derivative(y(x), x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-a*lambda_**2*sinh(lambda_*x) + a*y(x)**2*sinh(lambda_*x) + b*y(x)**2)/(a*sinh(lambda_*x) + b) cannot be solved by the factorable group method