50.1.59 problem 59

Internal problem ID [10057]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 59
Date solved : Tuesday, September 30, 2025 at 06:51:50 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 200

ode=D[y[x],x]-Sin[y[x]-x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.050 (sec). Leaf size: 15

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)
 

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