61.5.7 problem 7
Internal
problem
ID
[12131]
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
:
Tuesday, January 28, 2025 at 12:29:05 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*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 250
dsolve((a*sinh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*sinh(lambda*x)=0,y(x), singsol=all)
\[
y = -\frac {4 \left (\left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) a \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+a b \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \left (a^{2}+b^{2}\right )^{2}}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 a \cosh \left (\frac {\lambda x}{2}\right ) \left (a^{2}+b^{2}\right )^{{3}/{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 11.739 (sec). Leaf size: 229
DSolve[(a*Sinh[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)+a*\[Lambda]^2*Sinh[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {-a \sinh (\lambda K[1]) \lambda ^2+b y(x)^2+a \sinh (\lambda K[1]) y(x)^2}{(b+a \sinh (\lambda K[1])) (a \lambda \cosh (\lambda K[1])+b y(x)+a \sinh (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(a \lambda \cosh (x \lambda )+b K[2]+a K[2] \sinh (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (-a \sinh (\lambda K[1]) \lambda ^2+b K[2]^2+a K[2]^2 \sinh (\lambda K[1])\right )}{(a \lambda \cosh (\lambda K[1])+b K[2]+a K[2] \sinh (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \sinh (\lambda K[1]) K[2]}{(b+a \sinh (\lambda K[1])) (a \lambda \cosh (\lambda K[1])+b K[2]+a K[2] \sinh (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]