50.1.58 problem 58

Internal problem ID [10056]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 58
Date solved : Tuesday, September 30, 2025 at 06:51:08 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=-4 \sin \left (-y+x \right )-4 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 21

ode:=diff(y(x),x) = -4*sin(x-y(x))-4; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x +2 \arctan \left (\frac {3 \tan \left (-\frac {3 x}{2}+\frac {3 c_1}{2}\right )}{5}+\frac {4}{5}\right ) \]

✓ Mathematica. Time used: 0.698 (sec). Leaf size: 208

ode=D[y[x],x]==4*Sin[y[x]-x]-4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\frac {4 (\sin (K[1]-y(x))+1)}{4 \sin (K[1]-y(x))+5}dK[1]+\int _1^{y(x)}-\frac {4 \sin (x-K[2]) \int _1^x\left (\frac {16 \cos (K[1]-K[2]) (\sin (K[1]-K[2])+1)}{(4 \sin (K[1]-K[2])+5)^2}-\frac {4 \cos (K[1]-K[2])}{4 \sin (K[1]-K[2])+5}\right )dK[1]+5 \int _1^x\left (\frac {16 \cos (K[1]-K[2]) (\sin (K[1]-K[2])+1)}{(4 \sin (K[1]-K[2])+5)^2}-\frac {4 \cos (K[1]-K[2])}{4 \sin (K[1]-K[2])+5}\right )dK[1]-1}{4 \sin (x-K[2])+5}dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.891 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*sin(x - y(x)) + Derivative(y(x), x) + 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + x - \frac {2 \operatorname {atan}{\left (\frac {5 \tan {\left (\frac {x}{2} - \frac {y{\left (x \right )}}{2} \right )}}{3} + \frac {4}{3} \right )}}{3} - \frac {2 \pi \left \lfloor {\frac {x - y{\left (x \right )} - \pi }{2 \pi }}\right \rfloor }{3} = 0 \]