60.1.349 problem 350
Internal
problem
ID
[10363]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
350
Date
solved
:
Tuesday, January 28, 2025 at 04:36:16 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*}
✓ Solution by Maple
Time used: 0.592 (sec). Leaf size: 266
dsolve(diff(y(x),x)*cos(y(x))-cos(x)*sin(y(x))^2-sin(y(x)) = 0,y(x), singsol=all)
\begin{align*}
y &= \arctan \left (-\frac {2 \,{\mathrm e}^{x}}{\left (\cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{x}+2 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 \cos \left (x \right ) c_{1} {\mathrm e}^{x}+4 c_{1} \sin \left (x \right ) {\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 \,{\mathrm e}^{x}}{\left (\cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{x}+2 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 \cos \left (x \right ) c_{1} {\mathrm e}^{x}+4 c_{1} \sin \left (x \right ) {\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*}
✓ Solution by Mathematica
Time used: 0.416 (sec). Leaf size: 219
DSolve[-Sin[y[x]] - Cos[x]*Sin[y[x]]^2 + Cos[y[x]]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {1}{4} e^{K[1]} \csc (y(x)) (2 \cos (K[1]) \csc (y(x))-\cos (K[1]-2 y(x)) \csc (y(x))-\cos (K[1]+2 y(x)) \csc (y(x))+4)dK[1]+\int _1^{y(x)}\left (e^x \cot (K[2]) \csc (K[2])-\int _1^x\left (\frac {1}{4} e^{K[1]} \cot (K[2]) \csc (K[2]) (2 \cos (K[1]) \csc (K[2])-\cos (K[1]-2 K[2]) \csc (K[2])-\cos (K[1]+2 K[2]) \csc (K[2])+4)-\frac {1}{4} e^{K[1]} \csc (K[2]) (-2 \cos (K[1]) \cot (K[2]) \csc (K[2])+\cos (K[1]-2 K[2]) \cot (K[2]) \csc (K[2])+\cos (K[1]+2 K[2]) \cot (K[2]) \csc (K[2])-2 \sin (K[1]-2 K[2]) \csc (K[2])+2 \sin (K[1]+2 K[2]) \csc (K[2]))\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]