61.5.11 problem 11
Internal
problem
ID
[12135]
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
:
11
Date
solved
:
Tuesday, January 28, 2025 at 12:29:26 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 104
dsolve(diff(y(x),x)=(a*cosh(lambda*x)^2-lambda)*y(x)^2+a+lambda-a*cosh(lambda*x)^2,y(x), singsol=all)
\[
y = \frac {2 \tanh \left (\lambda x \right ) \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \left (a -\operatorname {sech}\left (\lambda x \right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {sech}\left (\lambda x \right )^{2} {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} c_{1} \lambda -\tanh \left (\lambda x \right )}{2 \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \left (a -\operatorname {sech}\left (\lambda x \right )^{2} \lambda \right )d x \right ) c_{1} -1}
\]
✓ Solution by Mathematica
Time used: 16.097 (sec). Leaf size: 211
DSolve[D[y[x],x]==(a*Cosh[\[Lambda]*x]^2-\[Lambda])*y[x]^2+a+\[Lambda]-a*Cosh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\text {sech}^2(\lambda x) \left (c_1 \sinh (2 \lambda x) \int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]+2 c_1 e^{\frac {a \cosh ^2(\lambda x)}{\lambda }}+\sinh (2 \lambda x)\right )}{2+2 c_1 \int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]} \\
y(x)\to \frac {1}{2} \text {sech}^2(\lambda x) \left (\frac {2 e^{\frac {a \cosh ^2(\lambda x)}{\lambda }}}{\int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]}+\sinh (2 \lambda x)\right ) \\
\end{align*}