36.1.1 problem 1

Internal problem ID [6256]
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
Date solved : Monday, January 27, 2025 at 01:50:13 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 25

dsolve(diff(y(x),x)-sin(x+y(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: 37.820 (sec). Leaf size: 541

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

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