32.6.28 problem Exercise 12.28, page 103

Internal problem ID [5893]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.28, page 103
Date solved : Monday, January 27, 2025 at 01:24:59 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 44

dsolve(x*diff(y(x),x)-x*sin(y(x)/x)-y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \arctan \left (\frac {2 x c_{1}}{c_{1}^{2} x^{2}+1}, \frac {-c_{1}^{2} x^{2}+1}{c_{1}^{2} x^{2}+1}\right ) x \]

✓ Solution by Mathematica

Time used: 0.329 (sec). Leaf size: 52

DSolve[x*D[y[x],x]-x*Sin[y[x]/x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x \arccos (-\tanh (\log (x)+c_1)) \\ y(x)\to x \arccos (-\tanh (\log (x)+c_1)) \\ y(x)\to 0 \\ y(x)\to -\pi x \\ y(x)\to \pi x \\ \end{align*}