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 = 0.26175 (sec), leaf count = 79
\[\left \{\left \{y(x)\to -2 \text {am}\left (\frac {1}{2} \sqrt {(2 a+c_1) (x+c_2){}^2}|\frac {4 a}{2 a+c_1}\right )\right \},\left \{y(x)\to 2 \text {am}\left (\frac {1}{2} \sqrt {(2 a+c_1) (x+c_2){}^2}|\frac {4 a}{2 a+c_1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.561 (sec), leaf count = 49
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right ) a +\textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right ) a +\textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] 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))
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]