61.16.1 problem 19
Internal
problem
ID
[12173]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.7-3.
Equations
containing
arctangent.
Problem
number
:
19
Date
solved
:
Wednesday, March 05, 2025 at 05:14:22 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+\lambda \arctan \left (x \right )^{n} y-a^{2}+a \lambda \arctan \left (x \right )^{n} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 71
ode:=diff(y(x),x) = y(x)^2+lambda*arctan(x)^n*y(x)-a^2+a*lambda*arctan(x)^n;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-c_{1} a -a \left (\int {\mathrm e}^{-\int \left (-\arctan \left (x \right )^{n} \lambda +2 a \right )d x}d x \right )-{\mathrm e}^{-\int \left (-\arctan \left (x \right )^{n} \lambda +2 a \right )d x}}{c_{1} +\int {\mathrm e}^{-\int \left (-\arctan \left (x \right )^{n} \lambda +2 a \right )d x}d x}
\]
✓ Mathematica. Time used: 1.104 (sec). Leaf size: 210
ode=D[y[x],x]==y[x]^2+\[Lambda]*ArcTan[x]^n*y[x]-a^2+a*\[Lambda]*ArcTan[x]^n;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right ) \left (-\lambda \arctan (K[2])^n+a-y(x)\right )}{n \lambda (a+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right )}{n \lambda (a+K[3])^2}-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right ) \left (-\lambda \arctan (K[2])^n+a-K[3]\right )}{n \lambda (a+K[3])^2}-\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right )}{n \lambda (a+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 63.455 (sec). Leaf size: 76
from sympy import *
x = symbols("x")
a = symbols("a")
cg = symbols("cg")
n = symbols("n")
y = Function("y")
ode = Eq(a**2 - a*cg*atan(x)**n - cg*y(x)*atan(x)**n - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\left (- C_{1} a e^{2 a x} + a e^{2 a x} \int e^{- 2 a x} e^{cg \int \operatorname {atan}^{n}{\left (x \right )}\, dx}\, dx + e^{cg \int \operatorname {atan}^{n}{\left (x \right )}\, dx}\right ) e^{- 2 a x}}{C_{1} - \int e^{- 2 a x} e^{cg \int \operatorname {atan}^{n}{\left (x \right )}\, dx}\, dx}
\]