56.1.59 problem 59

Internal problem ID [8771]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 59
Date solved : Monday, January 27, 2025 at 04:49:59 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.426 (sec). Leaf size: 261

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

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