47.2.9 problem 9

Internal problem ID [7425]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 9
Date solved : Monday, January 27, 2025 at 02:54:51 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y^{\prime }&=y \cos \left (\frac {y}{x}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 27

dsolve(x*diff(y(x),x)=y(x)*cos(y(x)/x),y(x), singsol=all)
 

\[ y = \operatorname {RootOf}\left (\ln \left (x \right )+c_{1} -\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \left (-1+\cos \left (\textit {\_a} \right )\right )}d \textit {\_a} \right ) x \]

✓ Solution by Mathematica

Time used: 2.237 (sec). Leaf size: 33

DSolve[x*D[y[x],x]==y[x]*Cos[y[x]/x],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{(\cos (K[1])-1) K[1]}dK[1]=\log (x)+c_1,y(x)\right ] \]