60.2.228 problem 804

Internal problem ID [10815]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 804
Date solved : Tuesday, January 28, 2025 at 05:16:46 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+\cos \left (2 y\right ) x^{4}+x^{4}}{2 x \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 38

dsolve(diff(y(x),x) = 1/2*(-sin(2*y(x))*x-sin(2*y(x))+cos(2*y(x))*x^4+x^4)/x/(x+1),y(x), singsol=all)
 

\[ y = \arctan \left (\frac {3 x^{4}-4 x^{3}+6 x^{2}+12 \ln \left (x +1\right )-12 c_{1} -12 x}{12 x}\right ) \]

✓ Solution by Mathematica

Time used: 0.395 (sec). Leaf size: 346

DSolve[D[y[x],x] == (x^4/2 + (x^4*Cos[2*y[x]])/2 - Sin[2*y[x]]/2 - (x*Sin[2*y[x]])/2)/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {1}{2} (\cos (2 y(x))+1) \sec ^2(y(x)) K[1]^3-\frac {1}{2} (\cos (2 y(x))+1) \sec ^2(y(x)) K[1]^2+\frac {1}{2} (\cos (2 y(x))+1) \sec ^2(y(x)) K[1]+\frac {\cos (2 y(x)) \sec ^2(y(x))+\sec ^2(y(x))}{2 (K[1]+1)}-\frac {1}{2} \sec ^2(y(x)) (\cos (2 y(x))+\sin (2 y(x))+1)\right )dK[1]+\int _1^{y(x)}\left (-x \sec ^2(K[2])-\int _1^x\left (-\sec ^2(K[2]) \sin (2 K[2]) K[1]^3+(\cos (2 K[2])+1) \sec ^2(K[2]) \tan (K[2]) K[1]^3+\sec ^2(K[2]) \sin (2 K[2]) K[1]^2-(\cos (2 K[2])+1) \sec ^2(K[2]) \tan (K[2]) K[1]^2-\sec ^2(K[2]) \sin (2 K[2]) K[1]+(\cos (2 K[2])+1) \sec ^2(K[2]) \tan (K[2]) K[1]-\frac {1}{2} \sec ^2(K[2]) (2 \cos (2 K[2])-2 \sin (2 K[2]))-\sec ^2(K[2]) (\cos (2 K[2])+\sin (2 K[2])+1) \tan (K[2])+\frac {-2 \sin (2 K[2]) \sec ^2(K[2])+2 \cos (2 K[2]) \tan (K[2]) \sec ^2(K[2])+2 \tan (K[2]) \sec ^2(K[2])}{2 (K[1]+1)}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]