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 : Sunday, March 30, 2025 at 10:44:51 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 25

ode:=diff(y(x),x)-sin(x+y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -x -2 \arctan \left (\frac {c_1 -x -2}{-x +c_1}\right ) \]

✓ Mathematica. Time used: 37.82 (sec). Leaf size: 541

ode=D[y[x],x]-Sin[x+y[x]]==0; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.715 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(x + y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - x - 2 \operatorname {atan}{\left (\frac {C_{1} + x + 2}{C_{1} + x} \right )} \]