Internal problem ID [4404]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page
46
Problem number: 1.
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: 25
dsolve(diff(y(x),x)-sin(x+y(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: 21.317 (sec). Leaf size: 501
DSolve[y'[x]-Sin[x+y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2 \text {ArcCos}\left (\frac {(x+c_1) \sin \left (\frac {x}{2}\right )-(x-2+c_1) \cos \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {(x-(1+i)+c_1) (x-(1-i)+c_1)}}\right ) \\ y(x)\to 2 \text {ArcCos}\left (\frac {(x+c_1) \sin \left (\frac {x}{2}\right )-(x-2+c_1) \cos \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {(x-(1+i)+c_1) (x-(1-i)+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-(1+i)+c_1) (x-(1-i)+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-(1+i)+c_1) (x-(1-i)+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*}