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 = 0.109613 (sec), leaf count = 40
\[\text {Solve}\left [\left \{x=\frac {c_1-(\text {K$\$$262842}-1) \text {K$\$$262842} \sin (\text {K$\$$262842})-\cos (\text {K$\$$262842})}{(\text {K$\$$262842}-1)^2},\text {K$\$$262842}^2 (\sin (\text {K$\$$262842})+x)=y(x)\right \},\{y(x),\text {K$\$$262842}\}\right ]\]
Maple ✓
cpu = 0.042 (sec), leaf count = 56
\[ \left \{ y \left ( x \right ) =0,[x \left ( {\it \_T} \right ) ={\frac { \left ( -{{\it \_T}}^{2}+{\it \_T} \right ) \sin \left ( {\it \_T} \right ) +{\it \_C1}-\cos \left ( {\it \_T} \right ) }{ \left ( {\it \_T}-1 \right ) ^{2}}},y \left ( {\it \_T} \right ) =-{\frac {{{\it \_T}}^{2} \left ( \left ( {\it \_T}-1 \right ) \sin \left ( {\it \_T} \right ) -{\it \_C1}+\cos \left ( {\it \_T} \right ) \right ) }{ \left ( {\it \_T}-1 \right ) ^{2}}}] \right \} \] 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$262842] - (-1 + K$262842)*K$262842*Sin[K$262842])/(-1 + K$262842)^2, K$262842^2*(x + Sin[K$262842]) == y[x]}, {y[x], K$262842}]
Maple raw input
dsolve(diff(y(x),x)^2*(x+sin(diff(y(x),x))) = y(x), y(x),'implicit')
Maple raw output
y(x) = 0, [x(_T) = ((-_T^2+_T)*sin(_T)+_C1-cos(_T))/(_T-1)^2, y(_T) = -_T^2*((_T-1)*sin(_T)-_C1+cos(_T))/(_T-1)^2]