61.6.2 problem 19
Internal
problem
ID
[12143]
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
:
19
Date
solved
:
Tuesday, January 28, 2025 at 12:34:14 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 148
dsolve(diff(y(x),x)=y(x)^2+3*a*lambda-lambda^2-a*(a+lambda)*tanh(lambda*x)^2,y(x), singsol=all)
\[
y = \frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda +2 \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) c_{1} \lambda -\tanh \left (\lambda x \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}
\]
✓ Solution by Mathematica
Time used: 6.911 (sec). Leaf size: 494
DSolve[D[y[x],x]==y[x]^2+3*a*\[Lambda]-\[Lambda]^2-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\int \frac {2 \lambda e^{-2 \lambda x} \left (\lambda \left (-2 e^{2 \lambda x}+e^{4 \lambda x}-1\right )-a \left (2 e^{2 \lambda x}+e^{4 \lambda x}-1\right )\right )}{e^{4 \lambda x}-1} \, de^{2 \lambda x}}{2 \lambda }-\frac {-2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {a (K[1]-1)^2-2 \lambda K[1] (K[1]+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)^2-2 \lambda K[1] (K[1]+1)}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]+6 \lambda x\right )+\left (\lambda \left (e^{2 \lambda x}+1\right )^2-a \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)^2-2 \lambda K[1] (K[1]+1)}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]\right )dK[2]+c_1 (\lambda -a)+c_1 (\lambda -a) e^{4 \lambda x}+2 c_1 (a+\lambda ) e^{2 \lambda x}}{\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)^2-2 \lambda K[1] (K[1]+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-\lambda \left (e^{2 \lambda x}+1\right )^2}{e^{4 \lambda x}-1}-\frac {\int \frac {2 \lambda e^{-2 \lambda x} \left (\lambda \left (-2 e^{2 \lambda x}+e^{4 \lambda x}-1\right )-a \left (2 e^{2 \lambda x}+e^{4 \lambda x}-1\right )\right )}{e^{4 \lambda x}-1} \, de^{2 \lambda x}}{2 \lambda } \\
\end{align*}