60.1.353 problem 354

Internal problem ID [10367]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 354
Date solved : Monday, January 27, 2025 at 07:33:09 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 108

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

\begin{align*} y &= \arctan \left (\frac {-\sqrt {c_{1}^{2}-x^{2}+1}\, c_{1} +x}{c_{1}^{2}+1}, \frac {c_{1} x +\sqrt {c_{1}^{2}-x^{2}+1}}{c_{1}^{2}+1}\right ) \\ y &= \arctan \left (\frac {\sqrt {c_{1}^{2}-x^{2}+1}\, c_{1} +x}{c_{1}^{2}+1}, \frac {c_{1} x -\sqrt {c_{1}^{2}-x^{2}+1}}{c_{1}^{2}+1}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.167 (sec). Leaf size: 163

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

\begin{align*} y(x)\to -\arccos \left (\frac {c_1 x-\sqrt {-x^2+1+c_1{}^2}}{1+c_1{}^2}\right ) \\ y(x)\to \arccos \left (\frac {c_1 x-\sqrt {-x^2+1+c_1{}^2}}{1+c_1{}^2}\right ) \\ y(x)\to -\arccos \left (\frac {\sqrt {-x^2+1+c_1{}^2}+c_1 x}{1+c_1{}^2}\right ) \\ y(x)\to \arccos \left (\frac {\sqrt {-x^2+1+c_1{}^2}+c_1 x}{1+c_1{}^2}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}