60.1.343 problem 350
Internal
problem
ID
[10357]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
350
Date
solved
:
Sunday, March 30, 2025 at 04:23:30 PM
CAS
classification
:
unknown
\begin{align*} y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right )&=0 \end{align*}
✓ Maple. Time used: 0.100 (sec). Leaf size: 262
ode:=diff(y(x),x)*cos(y(x))-cos(x)*sin(y(x))^2-sin(y(x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \arctan \left (-\frac {2}{\cos \left (x \right )+\sin \left (x \right )+2 \,{\mathrm e}^{-x} c_1}, \frac {\sqrt {\left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_1 \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_1 \,{\mathrm e}^{x}+4 c_1^{2}+{\mathrm e}^{2 x}\right ) \left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_1 \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_1 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+4 c_1^{2}\right )}}{2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_1 \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_1 \,{\mathrm e}^{x}+4 c_1^{2}+{\mathrm e}^{2 x}}\right ) \\
y &= \arctan \left (-\frac {2}{\cos \left (x \right )+\sin \left (x \right )+2 \,{\mathrm e}^{-x} c_1}, -\frac {\sqrt {\left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_1 \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_1 \,{\mathrm e}^{x}+4 c_1^{2}+{\mathrm e}^{2 x}\right ) \left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_1 \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_1 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+4 c_1^{2}\right )}}{2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_1 \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_1 \,{\mathrm e}^{x}+4 c_1^{2}+{\mathrm e}^{2 x}}\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.416 (sec). Leaf size: 219
ode=-Sin[y[x]] - Cos[x]*Sin[y[x]]^2 + Cos[y[x]]*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x-\frac {1}{4} e^{K[1]} \csc (y(x)) (2 \cos (K[1]) \csc (y(x))-\cos (K[1]-2 y(x)) \csc (y(x))-\cos (K[1]+2 y(x)) \csc (y(x))+4)dK[1]+\int _1^{y(x)}\left (e^x \cot (K[2]) \csc (K[2])-\int _1^x\left (\frac {1}{4} e^{K[1]} \cot (K[2]) \csc (K[2]) (2 \cos (K[1]) \csc (K[2])-\cos (K[1]-2 K[2]) \csc (K[2])-\cos (K[1]+2 K[2]) \csc (K[2])+4)-\frac {1}{4} e^{K[1]} \csc (K[2]) (-2 \cos (K[1]) \cot (K[2]) \csc (K[2])+\cos (K[1]-2 K[2]) \cot (K[2]) \csc (K[2])+\cos (K[1]+2 K[2]) \cot (K[2]) \csc (K[2])-2 \sin (K[1]-2 K[2]) \csc (K[2])+2 \sin (K[1]+2 K[2]) \csc (K[2]))\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 7.921 (sec). Leaf size: 53
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sin(y(x))**2*cos(x) - sin(y(x)) + cos(y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \operatorname {asin}{\left (\frac {2 e^{x}}{C_{1} + \sqrt {2} e^{x} \sin {\left (x + \frac {\pi }{4} \right )}} \right )} + \pi , \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {2 e^{x}}{C_{1} - \sqrt {2} e^{x} \sin {\left (x + \frac {\pi }{4} \right )}} \right )}\right ]
\]