55.6.2 problem 19
Internal
problem
ID
[13363]
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
:
Wednesday, October 01, 2025 at 08:35:31 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*}
✓ Maple. Time used: 0.080 (sec). Leaf size: 148
ode:=diff(y(x),x) = y(x)^2+3*a*lambda-lambda^2-a*(a+lambda)*tanh(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\left (a +\lambda \right ) \tanh \left (\lambda x \right ) \left (c_1 \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )\right )+2 c_1 \lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda }{c_1 \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}
\]
✓ Mathematica. Time used: 4.604 (sec). Leaf size: 494
ode=D[y[x],x]==y[x]^2+3*a*\[Lambda]-\[Lambda]^2-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-3*a*lambda_ + a*(a + lambda_)*tanh(lambda_*x)**2 + lambda_**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2*tanh(lambda_*x)**2 + a*lambda_*tanh(lambda_*x)**2 - 3*a*lambda_ + lambda_**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method