ODE
\[ a y'(x)^2+b \sin (y(x))+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.8647 (sec), leaf count = 146
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} c_1 a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} c_1+2 b \cos (K[1])}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} c_1 a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} c_1+2 b \cos (K[2])}}dK[2]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.763 (sec), leaf count = 126
\[\left [\int _{}^{y \left (x \right )}\frac {4 a^{2}+1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} \textit {\_C1} \,a^{2}-4 \sin \left (\textit {\_a} \right ) a b +2 b \cos \left (\textit {\_a} \right )+{\mathrm e}^{-2 a \textit {\_a}} \textit {\_C1} \right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {4 a^{2}+1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} \textit {\_C1} \,a^{2}-4 \sin \left (\textit {\_a} \right ) a b +2 b \cos \left (\textit {\_a} \right )+{\mathrm e}^{-2 a \textit {\_a}} \textit {\_C1} \right )}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[b*Sin[y[x]] + a*y'[x]^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][-(Sqrt[1 + 4*a^2]/Sqrt[C[1]/E^(2*a*K[1]) + (4*a^2*C[1])/E^(2*a*K[1]) + 2*b*Cos[K[1]] - 4*a*b*Sin[K[1]]]), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][Sqrt[1 + 4*a^2]/Sqrt[C[1]/E^(2*a*K[2]) + (4*a^2*C[1])/E^(2*a*K[2]) + 2*b*Cos[K[2]] - 4*a*b*Sin[K[2]]], {K[2], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*sin(y(x)) = 0, y(x))
Maple raw output
[Intat((4*a^2+1)/((4*a^2+1)*(4*exp(-2*a*_a)*_C1*a^2-4*sin(_a)*a*b+2*b*cos(_a)+exp(-2*a*_a)*_C1))^(1/2),_a = y(x))-x-_C2 = 0, Intat(-(4*a^2+1)/((4*a^2+1)*(4*exp(-2*a*_a)*_C1*a^2-4*sin(_a)*a*b+2*b*cos(_a)+exp(-2*a*_a)*_C1))^(1/2),_a = y(x))-x-_C2 = 0]