61.5.5 problem 5
Internal
problem
ID
[12129]
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
:
5
Date
solved
:
Tuesday, January 28, 2025 at 12:26:36 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 104
dsolve(diff(y(x),x)=(a*sinh(lambda*x)^2-lambda)*y(x)^2-a*sinh(lambda*x)^2+lambda-a,y(x), singsol=all)
\[
y = \frac {2 \coth \left (\lambda x \right ) \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \left (a -\operatorname {csch}\left (\lambda x \right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {csch}\left (\lambda x \right )^{2} {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} c_{1} \lambda -\coth \left (\lambda x \right )}{2 \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \left (a -\operatorname {csch}\left (\lambda x \right )^{2} \lambda \right )d x \right ) c_{1} -1}
\]
✓ Solution by Mathematica
Time used: 16.773 (sec). Leaf size: 211
DSolve[D[y[x],x]==(a*Sinh[\[Lambda]*x]^2-\[Lambda])*y[x]^2-a*Sinh[\[Lambda]*x]^2+\[Lambda]-a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\text {csch}^2(\lambda x) \left (c_1 \sinh (2 \lambda x) \int _1^xe^{\frac {a \sinh ^2(\lambda K[1])}{\lambda }} \text {csch}^2(\lambda K[1]) \left (\lambda -a \sinh ^2(\lambda K[1])\right )dK[1]+2 c_1 e^{\frac {a \sinh ^2(\lambda x)}{\lambda }}+\sinh (2 \lambda x)\right )}{2+2 c_1 \int _1^xe^{\frac {a \sinh ^2(\lambda K[1])}{\lambda }} \text {csch}^2(\lambda K[1]) \left (\lambda -a \sinh ^2(\lambda K[1])\right )dK[1]} \\
y(x)\to \frac {1}{2} \text {csch}^2(\lambda x) \left (\frac {2 e^{\frac {a \sinh ^2(\lambda x)}{\lambda }}}{\int _1^xe^{\frac {a \sinh ^2(\lambda K[1])}{\lambda }} \text {csch}^2(\lambda K[1]) \left (\lambda -a \sinh ^2(\lambda K[1])\right )dK[1]}+\sinh (2 \lambda x)\right ) \\
\end{align*}