ODE
\[ a \sin (y(x))+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0716378 (sec), leaf count = 79
\[\left \{\left \{y(x)\to -2 \text {am}\left (\frac {1}{2} \sqrt {\left (2 a+c_1\right ) \left (x+c_2\right ){}^2}|\frac {4 a}{2 a+c_1}\right )\right \},\left \{y(x)\to 2 \text {am}\left (\frac {1}{2} \sqrt {\left (2 a+c_1\right ) \left (x+c_2\right ){}^2}|\frac {4 a}{2 a+c_1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.107 (sec), leaf count = 49
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {2\,\cos \left ( {\it \_a} \right ) a+{\it \_C1}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt {2\,\cos \left ( {\it \_a} \right ) a+{\it \_C1}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[a*Sin[y[x]] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -2*JacobiAmplitude[Sqrt[(2*a + C[1])*(x + C[2])^2]/2, (4*a)/(2*a + C[1])]}, {y[x] -> 2*JacobiAmplitude[Sqrt[(2*a + C[1])*(x + C[2])^2]/2, (4*a)/(2*a + C[1])]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*sin(y(x)) = 0, y(x),'implicit')
Maple raw output
Intat(1/(2*cos(_a)*a+_C1)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(2*cos(_a)*a+_C1)^(1/2),_a = y(x))-x-_C2 = 0