60.1.122 problem 123

Internal problem ID [10136]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 123
Date solved : Monday, January 27, 2025 at 06:29:29 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.332 (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*}