83.3.7 problem 7

Internal problem ID [19063]
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. Exercise II (B) at page 9
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 12:49:52 PM
CAS classification : [_separable]

\begin{align*} \left (3+2 \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime }&=1+2 \sin \left (y\right )+\cos \left (y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 22

dsolve((3+2*sin(x)+cos(x))*diff(y(x),x)=1+2*sin(y(x))+cos(y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.390 (sec). Leaf size: 509

DSolve[(3+2*Sin[x]+Cos[x])*D[y[x],x]==1+2*Sin[y[x]]+Cos[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -2 \arccos \left (-\frac {2 e^{\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}}{\sqrt {-2 \exp \left (\arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )+2 c_1\right )+5 e^{2 \arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}+e^{2 \arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+4 c_1}}}\right ) \\ y(x)\to 2 \arccos \left (-\frac {2 e^{\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}}{\sqrt {-2 \exp \left (\arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )+2 c_1\right )+5 e^{2 \arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}+e^{2 \arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+4 c_1}}}\right ) \\ y(x)\to -2 \arccos \left (\frac {2 e^{\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}}{\sqrt {-2 \exp \left (\arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )+2 c_1\right )+5 e^{2 \arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}+e^{2 \arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+4 c_1}}}\right ) \\ y(x)\to 2 \arccos \left (\frac {2 e^{\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}}{\sqrt {-2 \exp \left (\arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )+2 c_1\right )+5 e^{2 \arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}+e^{2 \arctan \left (\tan \left (\frac {x}{2}\right )+1\right )+4 c_1}}}\right ) \\ \end{align*}