29.8.3 problem 208

Internal problem ID [4808]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 8
Problem number : 208
Date solved : Monday, January 27, 2025 at 09:40:27 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 44

dsolve(x*diff(y(x),x) = y(x)+x*sin(y(x)/x),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: 0.323 (sec). Leaf size: 52

DSolve[x D[y[x],x]==y[x]+x Sin[y[x]/x],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*}