75.12.15 problem 289

Internal problem ID [16881]
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 : Tuesday, January 28, 2025 at 09:39:20 AM
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*}

✓ Solution by Maple

Time used: 0.701 (sec). Leaf size: 62

dsolve(diff(y(x),x)+cos((x+y(x))/2)=cos((x-y(x))/2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.117 (sec). Leaf size: 76

DSolve[D[y[x],x]+Cos[(x+y[x])/2]==Cos[(x-y[x])/2],y[x],x,IncludeSingularSolutions -> True]
 

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