83.8.24 problem 25

Internal problem ID [19150]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 25
Date solved : Tuesday, January 28, 2025 at 01:06:49 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 21

dsolve(diff(y(x),x)=sin(x+y(x))+cos(x+y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.256 (sec). Leaf size: 387

DSolve[D[y[x],x]==Sin[x+y[x]]+Cos[x+y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -2 \arccos \left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )-\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\ y(x)\to 2 \arccos \left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )-\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\ y(x)\to -2 \arccos \left (\frac {-\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )+\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\ y(x)\to 2 \arccos \left (\frac {-\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )+\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\ \end{align*}