61.6.10 problem 27
Internal
problem
ID
[12151]
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
:
27
Date
solved
:
Tuesday, January 28, 2025 at 12:44:27 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 289
dsolve(diff(y(x),x)=y(x)^2+a*lambda+b*lambda-2*a*b-a*(a+lambda)*tanh(lambda*x)^2-b*(b+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[
y = \frac {-4 \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }} c_{1} \operatorname {csch}\left (\lambda x \right )^{2} \lambda \left (b -\frac {\lambda }{2}\right ) \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )-2 c_{1} \left (\left (\left (\frac {3 a}{2}+\frac {3 b}{2}\right ) \lambda +a b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }}+\left (-\frac {5 \,\operatorname {sech}\left (\lambda x \right ) \lambda \left (a +\frac {3 \lambda }{5}\right ) \operatorname {csch}\left (\lambda x \right )}{2}+a^{2} \tanh \left (\lambda x \right )\right ) \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (a \tanh \left (\lambda x \right )+\coth \left (\lambda x \right ) b \right ) \left (a +\frac {3 \lambda }{2}\right ) \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) c_{1} \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}+\left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )}
\]
✓ Solution by Mathematica
Time used: 22.769 (sec). Leaf size: 583
DSolve[D[y[x],x]==y[x]^2+a*\[Lambda]+b*\[Lambda]-2*a*b-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2-b*(b+\[Lambda])*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\int \frac {2 \lambda e^{-2 \lambda x} \left (a \left (2 e^{2 \lambda x}+4 e^{4 \lambda x}-6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )+b \left (-2 e^{2 \lambda x}+4 e^{4 \lambda x}+6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )\right )}{\left (e^{4 \lambda x}-1\right )^2} \, de^{2 \lambda x}}{2 \lambda }+\frac {2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {a (K[1]-1)^2+(K[1]+1) (K[1] b+b+\lambda -\lambda K[1])}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]+2 \lambda x\right )-2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {a (K[1]-1)^2+(K[1]+1) (K[1] b+b+\lambda -\lambda K[1])}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]+6 \lambda x\right )+\left (a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2\right ) \int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {a (K[1]-1)^2+(K[1]+1) (K[1] b+b+\lambda -\lambda K[1])}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]\right )dK[2]-2 c_1 (a-b) e^{2 \lambda x}+c_1 (a+b) e^{4 \lambda x}+c_1 (a+b)}{\left (e^{4 \lambda x}-1\right ) \left (\int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {a (K[1]-1)^2+(K[1]+1) (K[1] b+b+\lambda -\lambda K[1])}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]\right )dK[2]+c_1\right )} \\
y(x)\to \frac {a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2}{e^{4 \lambda x}-1}-\frac {\int \frac {2 \lambda e^{-2 \lambda x} \left (a \left (2 e^{2 \lambda x}+4 e^{4 \lambda x}-6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )+b \left (-2 e^{2 \lambda x}+4 e^{4 \lambda x}+6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )\right )}{\left (e^{4 \lambda x}-1\right )^2} \, de^{2 \lambda x}}{2 \lambda } \\
\end{align*}