67.3.9 problem 4.3 (i)

Internal problem ID [16330]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page 90
Problem number : 4.3 (i)
Date solved : Thursday, October 02, 2025 at 01:20:41 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

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

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

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

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 186

ode=D[y[x],x]==Sin[x+y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x-\exp \left (\int _1^{K[2]+y(x)}\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \sin (K[2]+y(x))dK[2]+\int _1^{y(x)}\left (\exp \left (\int _1^{x+K[3]}\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right )-\int _1^x\left (-\exp \left (\int _1^{K[2]+K[3]}\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \cos (K[2]+K[3])-\exp \left (\int _1^{K[2]+K[3]}\left (1-\frac {2}{\tan \left (\frac {K[1]}{2}\right )+1}\right )dK[1]\right ) \sin (K[2]+K[3]) \left (1-\frac {2}{\tan \left (\frac {1}{2} (K[2]+K[3])\right )+1}\right )\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.076 (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)
 

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