Internal problem ID [6350]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 59.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {y^{\prime }+\sin \left (x -y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.044 (sec). Leaf size: 23
dsolve(diff(y(x),x)-sin(y(x)-x)=0,y(x), singsol=all)
\[ y \relax (x ) = x +2 \arctan \left (\frac {c_{1}-x -2}{c_{1}-x}\right ) \]
✓ Solution by Mathematica
Time used: 19.682 (sec). Leaf size: 529
DSolve[y'[x]-Sin[y[x]-x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2 \text {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+c_1)}}\right ) \\ y(x)\to 2 \text {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+c_1)}}\right ) \\ y(x)\to -2 \text {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+c_1)}}\right ) \\ y(x)\to 2 \text {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+c_1)}}\right ) \\ y(x)\to -2 \text {ArcCos}\left (\frac {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}{\sqrt {2}}\right ) \\ y(x)\to 2 \text {ArcCos}\left (\frac {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}{\sqrt {2}}\right ) \\ y(x)\to -2 \text {ArcCos}\left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )}{\sqrt {2}}\right ) \\ y(x)\to 2 \text {ArcCos}\left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )}{\sqrt {2}}\right ) \\ y(x)\to -2 \text {ArcCos}\left (\frac {(x-2) \cos \left (\frac {x}{2}\right )-x \sin \left (\frac {x}{2}\right )}{\sqrt {2 (x-2) x+4}}\right ) \\ y(x)\to 2 \text {ArcCos}\left (\frac {(x-2) \cos \left (\frac {x}{2}\right )-x \sin \left (\frac {x}{2}\right )}{\sqrt {2 (x-2) x+4}}\right ) \\ y(x)\to -2 \text {ArcCos}\left (\frac {x \sin \left (\frac {x}{2}\right )-(x-2) \cos \left (\frac {x}{2}\right )}{\sqrt {2 (x-2) x+4}}\right ) \\ y(x)\to 2 \text {ArcCos}\left (\frac {x \sin \left (\frac {x}{2}\right )-(x-2) \cos \left (\frac {x}{2}\right )}{\sqrt {2 (x-2) x+4}}\right ) \\ \end{align*}