ODE
\[ y'(x)^2 \left (\sin \left (y'(x)\right )+x\right )=y(x) \] ODE Classification
[_dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.314432 (sec), leaf count = 48
\[\text {Solve}\left [\left \{x=\frac {-(K[1]-1) K[1] \sin (K[1])-\cos (K[1])+c_1}{(K[1]-1)^2},K[1]^2 (\sin (K[1])+x)=y(x)\right \},\{y(x),K[1]\}\right ]\]
Maple ✓
cpu = 2.701 (sec), leaf count = 68
\[\left [y \left (x \right ) = 0, \left [x \left (\textit {\_T} \right ) = \frac {-\textit {\_T}^{2} \sin \left (\textit {\_T} \right )-\cos \left (\textit {\_T} \right )+\textit {\_T} \sin \left (\textit {\_T} \right )+\textit {\_C1}}{\left (\textit {\_T} -1\right )^{2}}, y \left (\textit {\_T} \right ) = \frac {\left (-\textit {\_T}^{2} \sin \left (\textit {\_T} \right )-\cos \left (\textit {\_T} \right )+\textit {\_T} \sin \left (\textit {\_T} \right )+\textit {\_C1} \right ) \textit {\_T}^{2}}{\left (\textit {\_T} -1\right )^{2}}+\textit {\_T}^{2} \sin \left (\textit {\_T} \right )\right ]\right ]\] Mathematica raw input
DSolve[(x + Sin[y'[x]])*y'[x]^2 == y[x],y[x],x]
Mathematica raw output
Solve[{x == (C[1] - Cos[K[1]] - (-1 + K[1])*K[1]*Sin[K[1]])/(-1 + K[1])^2, K[1]^2*(x + Sin[K[1]]) == y[x]}, {y[x], K[1]}]
Maple raw input
dsolve(diff(y(x),x)^2*(x+sin(diff(y(x),x))) = y(x), y(x))
Maple raw output
[y(x) = 0, [x(_T) = 1/(_T-1)^2*(-_T^2*sin(_T)-cos(_T)+_T*sin(_T)+_C1), y(_T) = 1/(_T-1)^2*(-_T^2*sin(_T)-cos(_T)+_T*sin(_T)+_C1)*_T^2+_T^2*sin(_T)]]