60.1.196 problem 199

Internal problem ID [10210]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 199
Date solved : Sunday, March 30, 2025 at 03:31:04 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.399 (sec). Leaf size: 80

ode:=sin(2*x)*diff(y(x),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: 0.444 (sec). Leaf size: 68

ode=Sin[2*x]*D[y[x],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: 8.116 (sec). Leaf size: 80

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

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