61.6.1 problem 18
Internal
problem
ID
[12142]
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
:
18
Date
solved
:
Tuesday, January 28, 2025 at 12:34:09 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 122
dsolve(diff(y(x),x)=y(x)^2+a*lambda-a*(a+lambda)*tanh(lambda*x)^2,y(x), singsol=all)
\[
y = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{1} \lambda -\tanh \left (\lambda x \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )}
\]
✓ Solution by Mathematica
Time used: 5.214 (sec). Leaf size: 325
DSolve[D[y[x],x]==y[x]^2+a*\[Lambda]-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {a \int -\frac {\lambda \left (2-2 e^{-2 \lambda x}\right )}{e^{2 \lambda x}+1} \, de^{2 \lambda x}}{2 \lambda }+\frac {-2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {\frac {a (K[1]-1)}{\lambda (K[1]+1)}-1}{K[1]}dK[1]+2 \lambda x\right )-2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {\frac {a (K[1]-1)}{\lambda (K[1]+1)}-1}{K[1]}dK[1]+4 \lambda x\right )+a \left (e^{2 \lambda x}-1\right ) \int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {\frac {a (K[1]-1)}{\lambda (K[1]+1)}-1}{K[1]}dK[1]\right )dK[2]+a c_1 e^{2 \lambda x}-a c_1}{\left (e^{2 \lambda x}+1\right ) \left (\int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {\frac {a (K[1]-1)}{\lambda (K[1]+1)}-1}{K[1]}dK[1]\right )dK[2]+c_1\right )} \\
y(x)\to \frac {a \left (e^{2 \lambda x}-1\right )}{e^{2 \lambda x}+1}+\frac {a \int -\frac {\lambda \left (2-2 e^{-2 \lambda x}\right )}{e^{2 \lambda x}+1} \, de^{2 \lambda x}}{2 \lambda } \\
\end{align*}