29.5.10 problem 126

Internal problem ID [4727]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 5
Problem number : 126
Date solved : Sunday, March 30, 2025 at 03:50:24 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }+\csc \left (2 x \right ) \sin \left (2 y\right )&=0 \end{align*}

✓ Maple. Time used: 0.407 (sec). Leaf size: 80

ode:=diff(y(x),x)+csc(2*x)*sin(2*y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\arctan \left (-\frac {2 c_1 \sin \left (2 x \right )}{c_1^{2} \cos \left (2 x \right )-c_1^{2}-\cos \left (2 x \right )-1}, \frac {c_1^{2} \cos \left (2 x \right )-c_1^{2}+\cos \left (2 x \right )+1}{c_1^{2} \cos \left (2 x \right )-c_1^{2}-\cos \left (2 x \right )-1}\right )}{2} \]

✓ Mathematica. Time used: 0.438 (sec). Leaf size: 68

ode=D[y[x],x]+Csc[2 x] Sin[2 y[x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{2} \arccos (-\tanh (\text {arctanh}(\cos (2 x))+2 c_1)) \\ y(x)\to \frac {1}{2} \arccos (-\tanh (\text {arctanh}(\cos (2 x))+2 c_1)) \\ y(x)\to 0 \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}

✓ Sympy. Time used: 5.709 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + sin(2*y(x))/sin(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \pi - \frac {\operatorname {acos}{\left (\frac {- e^{4 C_{1}} \cos ^{2}{\left (x \right )} - \cos ^{2}{\left (x \right )} + 1}{e^{4 C_{1}} \cos ^{2}{\left (x \right )} - \cos ^{2}{\left (x \right )} + 1} \right )}}{2}, \ y{\left (x \right )} = \frac {\operatorname {acos}{\left (\frac {e^{4 C_{1}} \cos ^{2}{\left (x \right )} + \cos ^{2}{\left (x \right )} - 1}{- e^{4 C_{1}} \cos ^{2}{\left (x \right )} + \cos ^{2}{\left (x \right )} - 1} \right )}}{2}\right ] \]