60.1.340 problem 347
Internal
problem
ID
[10354]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
347
Date
solved
:
Sunday, March 30, 2025 at 04:23:08 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*}
✓ Maple. Time used: 0.142 (sec). Leaf size: 12
ode:=diff(y(x),x)*(sin(x)+1)*sin(y(x))+cos(x)*(cos(y(x))-1) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \arccos \left (\sin \left (x \right ) c_1 +c_1 +1\right )
\]
✓ Mathematica. Time used: 0.966 (sec). Leaf size: 381
ode=Cos[x]*(-1 + Cos[y[x]]) + (1 + Sin[x])*Sin[y[x]]*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 0.780 (sec). Leaf size: 27
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((sin(x) + 1)*sin(y(x))*Derivative(y(x), x) + (cos(y(x)) - 1)*cos(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \operatorname {acos}{\left (C_{1} \sin {\left (x \right )} + C_{1} + 1 \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (C_{1} \sin {\left (x \right )} + C_{1} + 1 \right )}\right ]
\]