69.12.15 problem 289

Internal problem ID [18168]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 289
Date solved : Thursday, October 02, 2025 at 03:06:39 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }+\cos \left (\frac {x}{2}+\frac {y}{2}\right )&=\cos \left (\frac {x}{2}-\frac {y}{2}\right ) \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 61

ode:=diff(y(x),x)+cos(1/2*x+1/2*y(x)) = cos(1/2*x-1/2*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 \arctan \left (\frac {2 c_1}{{\mathrm e}^{-2 \cos \left (\frac {x}{2}\right )} c_1^{2}+{\mathrm e}^{2 \cos \left (\frac {x}{2}\right )}}, \frac {-{\mathrm e}^{-4 \cos \left (\frac {x}{2}\right )} c_1^{2}+1}{{\mathrm e}^{-4 \cos \left (\frac {x}{2}\right )} c_1^{2}+1}\right ) \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 76

ode=D[y[x],x]+Cos[(x+y[x])/2]==Cos[(x-y[x])/2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-y(x) \int _1^x0dK[1]+\int _1^x-\left (\left (\cos \left (\frac {K[1]}{2}-\frac {y(x)}{2}\right )-\cos \left (\frac {K[1]}{2}+\frac {y(x)}{2}\right )\right ) \csc \left (\frac {y(x)}{2}\right )\right )dK[1]-2 \text {arctanh}\left (\cos \left (\frac {y(x)}{2}\right )\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.002 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-cos(x/2 - y(x)/2) + cos(x/2 + y(x)/2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - 2 \operatorname {acos}{\left (\frac {- C_{1} - e^{4 \cos {\left (\frac {x}{2} \right )}}}{C_{1} - e^{4 \cos {\left (\frac {x}{2} \right )}}} \right )} + 4 \pi , \ y{\left (x \right )} = 2 \operatorname {acos}{\left (\frac {- C_{1} - e^{4 \cos {\left (\frac {x}{2} \right )}}}{C_{1} - e^{4 \cos {\left (\frac {x}{2} \right )}}} \right )}\right ] \]