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 : Monday, January 27, 2025 at 09:34:29 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.451 (sec). Leaf size: 80

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]+Csc[2 x] Sin[2 y[x]]==0,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*}