56.1.59 problem 59

Internal problem ID [8771]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 59
Date solved : Sunday, March 30, 2025 at 01:32:48 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: 0.426 (sec). Leaf size: 261

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

\begin{align*} 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 {(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 ) \\ \end{align*}

✓ Sympy. Time used: 1.785 (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)
 

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