61.6.8 problem 25
Internal
problem
ID
[12149]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.4-2.
Equations
with
hyperbolic
tangent
and
cotangent.
Problem
number
:
25
Date
solved
:
Tuesday, January 28, 2025 at 07:19:40 PM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 302
dsolve((a*coth(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*coth(mu*x)*y(x)-d^2+c*d*coth(mu*x),y(x), singsol=all)
\[
y = \frac {-{\mathrm e}^{c \left (\int \frac {\coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x \right )} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (a \coth \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}-b^{2}\right )}}-d \left (\int \left (a \coth \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x \right )}d x -c_{1} \right )}{\int \left (a \coth \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x \right )}d x -c_{1}}
\]
✓ Solution by Mathematica
Time used: 38.678 (sec). Leaf size: 808
DSolve[(a*Coth[\[Lambda]*x]+b)*D[y[x],x]==y[x]^2+c*Coth[\[Mu]*x]*y[x]-d^2+c*d*Coth[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (d \cosh (\lambda K[2]-\mu K[2])-y(x) \cosh (\lambda K[2]-\mu K[2])-d \cosh (\lambda K[2]+\mu K[2])+c \sinh (\lambda K[2]-\mu K[2])+c \sinh (\lambda K[2]+\mu K[2])+\cosh (\lambda K[2]+\mu K[2]) y(x))}{c \mu (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (d \cosh (\lambda K[2]-\mu K[2])-K[3] \cosh (\lambda K[2]-\mu K[2])-d \cosh (\lambda K[2]+\mu K[2])+\cosh (\lambda K[2]+\mu K[2]) K[3]+c \sinh (\lambda K[2]-\mu K[2])+c \sinh (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3])^2 (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2]))}-\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (\cosh (\lambda K[2]+\mu K[2])-\cosh (\lambda K[2]-\mu K[2]))}{c \mu (d+K[3]) (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2]))}\right )dK[2]-\frac {e^{-\int _1^x\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]}}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ]
\]