61.5.17 problem 17
Internal
problem
ID
[12141]
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
:
17
Date
solved
:
Tuesday, January 28, 2025 at 12:34:03 AM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 204
dsolve((a*cosh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*cosh(lambda*x)=0,y(x), singsol=all)
\[
y = \frac {\left (-2 \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right ) a b \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right )+2 \sqrt {a^{2}-b^{2}}\, c_{1} a \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right )+\left (a -b \right ) \left (a +b \right ) \left (\cosh \left (\frac {\lambda x}{2}\right )^{2} a -\frac {a}{2}-\frac {b}{2}\right )\right ) \lambda }{\sqrt {a^{2}-b^{2}}\, \left (2 \left (\cosh \left (\frac {\lambda x}{2}\right )^{2} a -\frac {a}{2}+\frac {b}{2}\right ) b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )-a \sqrt {a^{2}-b^{2}}\, \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right )-2 c_{1} \left (\cosh \left (\frac {\lambda x}{2}\right )^{2} a -\frac {a}{2}+\frac {b}{2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 2.455 (sec). Leaf size: 229
DSolve[(a*Cosh[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)+a*\[Lambda]^2*Cosh[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {-a \cosh (\lambda K[1]) \lambda ^2+b y(x)^2+a \cosh (\lambda K[1]) y(x)^2}{(b+a \cosh (\lambda K[1])) (a \lambda \sinh (\lambda K[1])+b y(x)+a \cosh (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(b K[2]+a \cosh (x \lambda ) K[2]+a \lambda \sinh (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (-a \cosh (\lambda K[1]) \lambda ^2+b K[2]^2+a \cosh (\lambda K[1]) K[2]^2\right )}{(b K[2]+a \cosh (\lambda K[1]) K[2]+a \lambda \sinh (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \cosh (\lambda K[1]) K[2]}{(b+a \cosh (\lambda K[1])) (b K[2]+a \cosh (\lambda K[1]) K[2]+a \lambda \sinh (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]