60.1.346 problem 347

Internal problem ID [10360]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 347
Date solved : Monday, January 27, 2025 at 07:30:33 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.230 (sec). Leaf size: 12

dsolve(diff(y(x),x)*(1+sin(x))*sin(y(x))+cos(x)*(cos(y(x))-1) = 0,y(x), singsol=all)
 

\[ y = \arccos \left (\sin \left (x \right ) c_{1} +c_{1} +1\right ) \]

✓ Solution by Mathematica

Time used: 0.966 (sec). Leaf size: 381

DSolve[Cos[x]*(-1 + Cos[y[x]]) + (1 + Sin[x])*Sin[y[x]]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 0 \\ \text {Solve}\left [\int _1^x\exp \left (\int _1^{K[2]}\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \left (\cos \left (\frac {K[2]}{2}-\frac {y(x)}{2}\right )-\cos \left (\frac {K[2]}{2}+\frac {y(x)}{2}\right )+\sin \left (\frac {K[2]}{2}-\frac {y(x)}{2}\right )-\sin \left (\frac {K[2]}{2}+\frac {y(x)}{2}\right )\right )dK[2]+\int _1^{y(x)}\left (\exp \left (\int _1^x\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \cos \left (\frac {x}{2}-\frac {K[3]}{2}\right )+\exp \left (\int _1^x\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \cos \left (\frac {x}{2}+\frac {K[3]}{2}\right )+\exp \left (\int _1^x\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \sin \left (\frac {x}{2}-\frac {K[3]}{2}\right )+\exp \left (\int _1^x\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \sin \left (\frac {x}{2}+\frac {K[3]}{2}\right )-\int _1^x\exp \left (\int _1^{K[2]}\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \left (-\frac {1}{2} \cos \left (\frac {K[2]}{2}-\frac {K[3]}{2}\right )-\frac {1}{2} \cos \left (\frac {K[2]}{2}+\frac {K[3]}{2}\right )+\frac {1}{2} \sin \left (\frac {K[2]}{2}-\frac {K[3]}{2}\right )+\frac {1}{2} \sin \left (\frac {K[2]}{2}+\frac {K[3]}{2}\right )\right )dK[2]\right )dK[3]&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}