61.5.14 problem 14
Internal
problem
ID
[12138]
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
:
14
Date
solved
:
Tuesday, January 28, 2025 at 12:30:46 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 246
dsolve(diff(y(x),x)=a*sinh(lambda*x)*y(x)^2+b*sinh(lambda*x)*cosh(lambda*x)^n,y(x), singsol=all)
\[
y = -\frac {\left (\lambda \sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )-a \cosh \left (\lambda x \right )^{\frac {n}{2}+1} \left (\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) \sqrt {b}\right ) \operatorname {sech}\left (\lambda x \right )}{a^{{3}/{2}} \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.716 (sec). Leaf size: 667
DSolve[D[y[x],x]==a*Sinh[\[Lambda]*x]*y[x]^2+b*Sinh[\[Lambda]*x]*Cosh[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cosh ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\text {sech}(\lambda x) \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \left (\sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )+\lambda \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )+\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cosh ^{\frac {n}{2}+1}(\lambda x) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{2 a \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}-\lambda \text {sech}(\lambda x)}{2 a} \\
\end{align*}