60.2.230 problem 806

Internal problem ID [10817]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 806
Date solved : Tuesday, January 28, 2025 at 05:17:13 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 22

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

\[ y = -\arctan \left (\frac {\ln \left (x +1\right )-x -c_{1}}{x}\right ) \]

✓ Solution by Mathematica

Time used: 0.316 (sec). Leaf size: 190

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

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