75.4.30 problem 95

Internal problem ID [16661]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 95
Date solved : Monday, March 31, 2025 at 03:04:01 PM
CAS classification : [_separable]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }-\frac {\cos \left (2 y\right )^{2}}{2}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (-\infty \right )&=\frac {7 \pi }{2} \end{align*}

✓ Maple. Time used: 0.301 (sec). Leaf size: 17

ode:=(x^2+1)*diff(y(x),x)-1/2*cos(2*y(x))^2 = 0; 
ic:=y(-infinity) = 7/2*Pi; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\arctan \left (\arctan \left (x \right )+\frac {\pi }{2}\right )}{2}+\frac {7 \pi }{2} \]

✗ Mathematica

ode=(1+x^2)*D[y[x],x]-1/2*Cos[2*y[x]]^2==0; 
ic={y[-Infinity]==7/2*Pi}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 1)*Derivative(y(x), x) - cos(2*y(x))**2/2,0) 
ics = {y(-inf): 7*pi/2} 
dsolve(ode,func=y(x),ics=ics)